Auction and negotiation algorithms for cooperative task allocation

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Content AUCTIONANDNEGOTIATIONALGORITHMSFORCOOPERATIVETASK ALLOCATION by XiaomingZheng ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (COMPUTERSCIENCE) December2014 Copyright 2014 XiaomingZheng Abstract It is often important to coordinate a team of cooperative robots well in a distributed computing environment. In this dissertation, we study how to allocate and re-allocate tasks to distributed robots so that the team cost is as small as possible (= the team per- formance is as high as possible). Researchers have developed several algorithms based on auction-like and negotiation-like protocols for decentralized task allocation. How- ever, the majority of these existing algorithms use either single-item auctions, in which only one task is allocated to some robot in one round so that the team cost increases the least, or single-item exchanges, in which only one task is transferred between two robots in one round so that the team cost decreases the most. These algorithms usually result in highly sub-optimal allocations and do not apply to complex tasks that need to beexecutedbymorethanonerobotsimultaneously. • Wedevelopanewauctionalgorithm,calledsequentialauctionswithbundles,that extends single-item auctions to be able to allocate more than one task to robots in one round so that the team cost increases the least. We introduce a novel data structure, called bid trees, that each robot can construct and submit to the auc- tioneer independently. Theoretical results show that the bids from bid trees can succinctly characterize all necessary local information of robots needed by the auctioneer to allocate multiple tasks to robots in one round so that the team cost ii increases the least. Experimental results show that sequential auctions with bun- dlesreducetheteamcostsofsingle-itemauctionssignificantly. • We develop a new negotiation algorithm, called sequential negotiations with K- swaps,thatextendssingle-itemexchangestobeabletore-allocatemorethanone task among robots in one round so that the team cost decreases the most. We introduce a novel data structure, called partialk-swaps, that each robot can con- struct and propose to other robots independently. Theoretical results show that profitablepartialk-swapscansuccinctlycharacterizeallnecessarylocalinforma- tion of robots needed to re-allocate multiple tasks among them so that the team cost decreases the most. Experimental results show that sequential negotiations withK-swapsreducetheteamcostsofgiveninitialallocationssignificantly. • We develop a new auction algorithm, called sequential auctions with reaction functions, that extends single-item auctions to be able to allocate either a simple or complex task to robots in one round so that the team cost increases the least. Weintroduceanoveldatastructure,calledreactionfunctions,thateachrobotcan construct and submit to the auctioneer independently. Theoretical results show thatreactionfunctionscansuccinctlycharacterizeallnecessarylocalinformation of robots needed by the auctioneer to allocate either a simple or complex task to robotsinoneroundsothattheteamcostincreasestheleast. Experimentalresults show that sequential auctions with reaction functions reduce the team costs of an existing auction algorithm developed by Shehory and Kraus [SK95, SK98] significantly. • Finally, we develop a new negotiation algorithm, called sequential negotiations withreactionfunctions,thatextendssingle-itemexchangestobeabletore-allocate complexorsimpletasksamongrobotsinoneroundsothattheteamcostdecreases iii the most. Theoretical results show that reaction functions can succinctly charac- terize all necessary local information of robots needed to re-allocate complex or simple tasks among them so that the team cost decreases the most. Experimental results show that sequential negotiations with reaction functions reduce the team costsofgiveninitialallocationssignificantly. To summarize, in this dissertation we develop new auction and negotiation algo- rithmsforsolvingtask-allocationproblemswithsimpleandcomplextasksanddemon- strate empirically that these new algorithms reduce the team costs of existing ones sig- nificantly. iv Acknowledgments Firstandforemost,Iwouldliketothankmyadvisor,SvenKoenig,forhishelpthrough- outthisjourney. Ithankmycommitteemembersfortheirtimeandguidance. Inpartic- ular, it has been a great pleasure to collaborate with Craig Tovey for manyencouraging conversations. Iamgratefultohim,DavidKempe,WeiMinShen,andMagedDessouky. I would also like to thank my colleagues and collaborators Xiaoxun Sun, William Yeoh, Po-An Chen, Kenny Daniel, Alex Nash, Janusz Marecki for all the stimulating discussions we have had. Last but not least, I would like to thank my parents and wife forbeingsounderstandingandpatientwithme. v Contents Abstract ii Acknowledgments v ListofFigures ix ListofTables xiii 1 Introduction 1 1.1 Task-AllocationProblem . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 MotivatingDomain: Multi-RobotRouting . . . . . . . . . . . . 3 1.1.2 TaskAllocationinDistributedProblemSolving . . . . . . . . 6 1.1.3 EvaluationCriteria . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 SequentialSingle-ItemAuctions . . . . . . . . . . . . . . . . . . . . . 9 1.3 OurApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 SystemFrameworks . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 PerformancePrinciples . . . . . . . . . . . . . . . . . . . . . . 24 1.4 OurContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5 DissertationStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Background 33 2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.1 ApplicationDomains . . . . . . . . . . . . . . . . . . . . . . . 34 2.1.2 RobotModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.3 TaskModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.4 TeamObjectives . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.1 CentralizedApproaches . . . . . . . . . . . . . . . . . . . . . 43 2.2.2 DistributedApproaches . . . . . . . . . . . . . . . . . . . . . 46 2.2.3 Market-BasedApproaches . . . . . . . . . . . . . . . . . . . . 50 vi 3 Simple-TaskAuctions 59 3.1 ProblemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 ExistingApproachestoSimple-TaskAuctions . . . . . . . . . . . . . 63 3.2.1 CombinatorialAuctions . . . . . . . . . . . . . . . . . . . . . 63 3.2.2 ParallelAuctions . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.3 SequentialSingle-ItemAuctions . . . . . . . . . . . . . . . . . 67 3.3 SequentialAuctionswithBundles . . . . . . . . . . . . . . . . . . . . 70 3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.2 Bid-ConstructionPhase . . . . . . . . . . . . . . . . . . . . . 77 3.3.3 Winner-DeterminationPhase . . . . . . . . . . . . . . . . . . 82 3.3.4 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4 ExperimentalEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4.1 ExperimentalSettings . . . . . . . . . . . . . . . . . . . . . . 87 3.4.2 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . 93 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4 Simple-TaskNegotiations 96 4.1 ProblemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 ExistingApproachestoSimple-TaskNegotiations . . . . . . . . . . . 99 4.3 SequentialNegotiationswithK-Swaps . . . . . . . . . . . . . . . . . . 103 4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.2 Partialk-Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.3 NegotiationAlgorithm . . . . . . . . . . . . . . . . . . . . . . 118 4.4 ExperimentalEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4.1 ExperimentalSettings . . . . . . . . . . . . . . . . . . . . . . 122 4.4.2 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . 123 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5 Complex-TaskAuctions 127 5.1 ProblemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2 ExistingApproachestoComplexTaskAuctions . . . . . . . . . . . . . 131 5.3 DisjointCoalitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.2 ReactionFunctions . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3.3 SequentialAuctionswithReactionFunctions . . . . . . . . . . 150 5.4 OverlappingCoalitions . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.4.2 GeneralizedReactionFunctions . . . . . . . . . . . . . . . . . 157 5.4.3 SequentialAuctionswithGeneralizedReactionFunctions . . . 174 5.5 ExperimentalEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.5.1 ExperimentalSettings . . . . . . . . . . . . . . . . . . . . . . 178 5.5.2 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . 181 vii 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6 Complex-TaskNegotiations 184 6.1 ProblemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.2 DisjointCoalitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2.1 Complex-TaskExchanges . . . . . . . . . . . . . . . . . . . . 188 6.2.2 Simple-TaskExchanges . . . . . . . . . . . . . . . . . . . . . 198 6.2.3 SequentialNegotiationswithReactionFunctions . . . . . . . . 207 6.3 OverlappingCoalitions . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.3.1 Complex-TaskExchanges . . . . . . . . . . . . . . . . . . . . 209 6.3.2 Simple-TaskExchanges . . . . . . . . . . . . . . . . . . . . . 217 6.3.3 SequentialNegotiationswithGeneralizedReactionFunctions . 224 6.4 ExperimentalEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.4.1 ExperimentalSettings . . . . . . . . . . . . . . . . . . . . . . 225 6.4.2 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . 227 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7 Conclusions 230 ReferenceList 235 A ExperimentalDataandExplanations 266 A.1 Simple-TaskAuctionExperiments . . . . . . . . . . . . . . . . . . . . 266 A.2 Simple-TaskNegotiationExperiments . . . . . . . . . . . . . . . . . . 295 A.3 Complex-TaskAuctionExperiments . . . . . . . . . . . . . . . . . . . 323 A.4 Complex-TaskNegotiationExperiments . . . . . . . . . . . . . . . . . 370 viii ListofFigures 1.1 Example1ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 4 1.2 AGeneralModelofDistributedProblemSolving . . . . . . . . . . . . 6 1.3 TwoStagesofTask-AllocationPhase . . . . . . . . . . . . . . . . . . . 8 1.4 InteractionDiagramofContractNetProtocol . . . . . . . . . . . . . . 15 1.5 InteractionDiagramofIterativeAuctionProtocol . . . . . . . . . . . . 17 1.6 Example2ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . . 17 1.7 ExampleofIterativeAuctionProtocol . . . . . . . . . . . . . . . . . . 17 1.8 InteractionDiagramofOne-to-OneNegotiationProtocol . . . . . . . . 19 1.9 ExampleofOne-to-OneNegotiationProtocol . . . . . . . . . . . . . . 20 1.10 InteractionDiagramofMulti-PartyNegotiationProtocol . . . . . . . . 21 1.11 Example3ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . . 22 1.12 ExampleofMulti-PartyNegotiationProtocol . . . . . . . . . . . . . . 22 3.1 Example4ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 61 3.2 AllocationsforExampleinFigure3.1(MiniMaxTeamObjective) . . . 64 3.3 ConnectionbetweenCombinatorialAuctionsandSequentialSingle-Item Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Connection between Sequential Auctions with Bundles and Existing AuctionAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 BidTreesforSequentialAuctionswithBundleSizeTwo . . . . . . . . 81 3.6 BidTreesforSequentialAuctionswithBundleSizeThree . . . . . . . 82 ix 3.7 SearchTreesforSequentialAuctionswithBidTreesandBundleSizeTwo 84 3.8 Search Trees for Sequential Auctions with Bid Trees and Bundle Size Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.9 Allocations of Sequential Auctions with Bundle Size Two and Refine- mentforExampleinFigure3.1(MiniMaxTeamObjective) . . . . . . 88 3.10 Experimental Test-Beds for Multi-Robot Routing: Outdoor-Like Ter- rain,EmptyTerrain,andIndoor-LikeTerrain . . . . . . . . . . . . . . 90 4.1 Example6ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 98 4.2 AllocationsforExampleinFigure4.1(MiniMaxTeamObjective) . . . 99 4.3 Connectionbetweenk-SwapsandExistingNegotiationAlgorithms . . 104 4.4 OutSwap,InSwapandExchangeSwapforExampleinFigure4.1 . . . 106 4.5 PartialandCompletek-SwapforExampleinFigure4.1(MiniMaxTeam Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.6 RemovingRobotfromPartialk-Swap . . . . . . . . . . . . . . . . . . 111 4.7 Example7ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . . 115 5.1 Example9ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 134 5.2 AllocationsforExampleinFigure5.1(MiniMaxTeamObjective) . . . 134 5.3 Meta-Functions of Robot r 1 for Complex Target x 5 (MiniMax Team Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Meta-Functions of Robot r 2 for Complex Target x 5 (MiniMax Team Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.5 Reaction Functions of Robots for Complex Target x 5 (MiniMax Team Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.6 Or-OptApproximationAlgorithm . . . . . . . . . . . . . . . . . . . . 148 5.7 ApproximateReactionFunctionsofRobotsforComplexTargetx 5 (Min- iMaxTeamObjective) . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.8 Example10ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 157 x 5.9 DomainsofGeneralizedReactionFunctionofRobotr 2 (MiniMaxTeam Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.10 DomainBoundariesofOneMeta-FunctionofRobotr 2 (MiniMaxTeam Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.11 Domainsof(Generalized)ReactionFunctionsofRobots(MiniMaxTeam Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.12 SearchTreeforDeterminingOptimalVisitTimes(MiniMaxTeamObjec- tive) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.13 DomainBoundariesforDeterminingOptimalVisitTimes . . . . . . . 170 5.14 Approximating Generalized Reaction Function of Robot r 2 (MiniMax TeamObjective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.1 Example12ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 186 6.2 AllocationsofExampleinFigure6.1(MiniMaxTeamObjective) . . . 186 6.3 Reaction Functions of Robot r 1 for Sequential Auctions with Reaction FunctionsforExampleinFigure6.1(MiniMaxTeamObjective) . . . . 188 6.4 Reaction Functions of Robot r 2 for Sequential Auctions with Reaction FunctionsforExampleinFigure6.1(MiniMaxTeamObjective) . . . . 188 6.5 Reaction Functions of Robot r 3 for Sequential Auctions with Reaction FunctionsforExampleinFigure6.1(MiniMaxTeamObjective) . . . . 189 6.6 Reaction Functions of Robot r 4 for Sequential Auctions with Reaction FunctionsforExampleinFigure6.1(MiniMaxTeamObjective) . . . . 189 6.7 AllocationsafterComplex-TaskExchange(r 1 ,r 3 ,x 3 ,x 2 )(MiniMaxTeam Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.8 Reaction Functions of Robots r 1 and r 2 after Simple-Task Exchanges forExampleinFigure6.1(MiniMaxTeamObjective) . . . . . . . . . 200 6.9 Reaction Functions of Robots r 3 and r 4 after Simple-Task Exchanges forExampleinFigure6.1(MiniMaxTeamObjective) . . . . . . . . . 200 A.1 Example5ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 278 A.2 AllocationsforExampleinFigureA.1(MiniMaxTeamObjective) . . . 278 A.3 Example8ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 311 xi A.4 AllocationsforExampleinFigureA.3(MiniMaxTeamObjective) . . . 311 A.5 Example11ofMulti-RobotRouting . . . . . . . . . . . . . . . . . . . 347 xii ListofTables 1.1 SystemFrameworksofOurTask-AllocationAlgorithms . . . . . . . . 23 1.2 PerformancePrinciplesofOurTask-AllocationAlgorithms . . . . . . 24 1.3 CategoriesofOurTask-AllocationAlgorithms . . . . . . . . . . . . . 24 1.4 TheoreticalContributionsofOurTask-AllocationAlgorithms . . . . . 28 3.1 RobotCostsforExampleinFigure3.1(MiniMaxTeamObjective) . . 61 3.2 Combinatorial Auctions for Example in Figure 3.1 (MiniMax Team Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3 ParallelAuctionsforExampleinFigure3.1(MiniMaxTeamObjective) 66 3.4 Sequential Single-Item Auctions for Example in Figure 3.1 (MiniMax TeamObjective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Sequential Auctions with Bundle Size Two for Example in Figure 3.1 (MiniMaxTeamObjective) . . . . . . . . . . . . . . . . . . . . . . . 75 3.6 UnnecessaryBidsSubmittedbyRobotsinTable3.5 . . . . . . . . . . 78 3.7 ReductioninNumberofBidsSubmitted . . . . . . . . . . . . . . . . . 79 3.8 Sequential Auctions with Bid Trees and Bundle Size Two for Example inFigure3.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . . . 86 3.9 SequentialAuctionswithBundleSizeTwoandRefinementforExample inFigure3.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . . . 87 3.10 SummaryofSimple-TaskAuctionExperiments . . . . . . . . . . . . . 93 4.1 RobotCostsforExampleinFigure4.1(MiniMaxTeamObjective) . . . 98 4.2 SingleTaskExchangesforExampleinFigure4.1(MiniMaxTeamObjec- tive) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 xiii 4.3 Combinatorial Exchanges for Example in Figure 4.1 (MiniMax Team Objective)(Part1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4 Combinatorial Exchanges for Example in Figure 4.1 (MiniMax Team Objective)(Part2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5 Partial Zero-Swaps Sent for Example in Figure 4.1 (MiniMax Team Objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.6 SummaryofSimple-TaskNegotiationExperiments . . . . . . . . . . . 124 5.1 SKAuctionsforExampleinFigure5.1(MiniMaxTeamObjective) . . 135 5.2 VisitOrdersofRobotsforExampleinFigure5.1 . . . . . . . . . . . . 141 5.3 VisitOrdersofRobotr 2 forExampleinFigure5.8 . . . . . . . . . . . 159 5.4 PermutationsandScenariosforExampleinFigure5.8 . . . . . . . . . 167 5.5 SummaryofComplex-TaskAuctionExperiments . . . . . . . . . . . . 182 6.1 SequentialAuctionswithReactionFunctionsforExampleinFigure6.1 (MiniMaxTeamObjective) . . . . . . . . . . . . . . . . . . . . . . . 187 6.2 Task-ExchangeEvaluationPhaseofComplex-TaskExchangesforExam- pleinFigure6.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . 192 6.3 Proposal-EvaluationPhaseofComplex-TaskExchangesforExamplein Figure6.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . . . . . 193 6.4 Task-ExchangeEvaluationPhaseofSimple-TaskExchangesforExam- pleinFigure6.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . 201 6.5 Proposal-Evaluation Phase of Simple-Task Exchanges for Example in Figure6.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . . . . . 202 6.6 SummaryofComplex-TaskNegotiationExperiments . . . . . . . . . . 228 A.1 Simple-TaskAuctions: ComparisonofTeamCostswithOptimalAlgo- rithminOutdoor-LikeTerrain . . . . . . . . . . . . . . . . . . . . . . 269 A.2 Simple-TaskAuctions: ComparisonofTeamCostsinOutdoor-LikeTer- rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 A.3 Simple-Task Auctions: Comparison of Response Times in Outdoor- LikeTerrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 xiv A.4 Simple-TaskAuctions: ComparisonofSABAlgorithmswithandwith- outRefinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 A.5 Simple-TaskAuctions: ComparisonofTeamCostsinEmptyTerrain . 271 A.6 Simple-Task Auctions: Comparison of Response Times in Empty Ter- rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 A.7 Simple-Task Auctions: Comparison of Team Costs in Office-Like Ter- rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 A.8 Simple-Task Auctions: Comparison of Response Times in Office-Like Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 A.9 Simple-Task Auctions: Comparison of Team Costs for Different Robot Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 A.10 Simple-Task Auctions: Comparison of Response Times for Different RobotCapacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 A.11 Simple-TaskAuctions: ComparisonofTeamCostsforIdealRobot-Cost Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 A.12 Simple-TaskAuctions: ComparisonofResponseTimesforIdealRobot- CostCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 A.13 Simple-Task Auctions: Comparison of Team Costs for Loose Robot- CapacityConstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 A.14 Simple-TaskAuctions: ComparisonofResponseTimesforLooseRobot- CapacityConstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 A.15 Simple-TaskAuctions: ComparisonofTeamCostsinUnknownTerrain 276 A.16 Simple-Task Auctions: Comparison of Numbers of Repeated Execu- tionsinUnknownTerrain . . . . . . . . . . . . . . . . . . . . . . . . . 276 A.17 Simple-Task Auctions: Comparison of Cumulative Response Times in UnknownTerrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 A.18 ComparisonofParallelAuctionsandSAB(k=1)forExampleinFigure A.1(MiniMaxTeamObjective) . . . . . . . . . . . . . . . . . . . . . 279 A.19 ComparisonofResponseTimesforSAB(k=3)withandwithoutRefine- ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 xv A.20 Number of Targets Allocated to Robots with Loose Robot-Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 A.21 Sequential Auctions with Bundle Size Two for Example in A.1 (Mini- MaxTeamObjective) . . . . . . . . . . . . . . . . . . . . . . . . . . 299 A.22 Simple-Task Negotiations: Comparison of Team Costs with Optimal AlgorithminOutdoor-LikeTerrain . . . . . . . . . . . . . . . . . . . 300 A.23 Simple-TaskNegotiations: ComparisonofTeamCostsinOutdoor-Like Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 A.24 Simple-Task Negotiations: Comparison of Amounts of Robot Commu- nicationforSNW(K =1,2)Algorithms . . . . . . . . . . . . . . . . . 301 A.25 Simple-Task Negotiations: Comparison of Amounts of Robot Commu- nicationforSNW(K =3)Algorithm . . . . . . . . . . . . . . . . . . . 302 A.26 Simple-TaskNegotiations: ComparisonofResponseTimesinOutdoor- LikeTerrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 A.27 Simple-TaskNegotiations: ComparisonofTeamCostsinEmptyTerrain 303 A.28 Simple-Task Negotiations: Comparison of Response Times in Empty Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 A.29 Simple-Task Negotiations: Comparison of Team Costs in Indoor-Like Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 A.30 Simple-Task Negotiations: Comparison of Response Times in Indoor- LikeTerrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 A.31 Simple-Task Negotiations: Comparison of Team Costs for Different RobotCapacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 A.32 Simple-Task Negotiations: Comparison of Response Times for Differ- entRobotCapacities . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 A.33 Simple-TaskNegotiations: ComparisonofTeamCostsforIdealRobot- CostCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 A.34 Simple-Task Negotiations: Comparison of Response Times for Ideal Robot-CostCalculations . . . . . . . . . . . . . . . . . . . . . . . . . 306 A.35 Simple-TaskNegotiations: ComparisonofTeamCostsforRandomIni- tialAllocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 xvi A.36 Simple-TaskNegotiations: ComparisonofResponseTimesforRandom InitialAllocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 A.37 Simple-TaskNegotiations: ComparisonofTeamCostsinUnknownTer- rain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 A.38 Simple-Task Negotiations: Comparison of Numbers of Repeated Exe- cutionsinUnknownTerrain . . . . . . . . . . . . . . . . . . . . . . . 308 A.39 Simple-TaskNegotiations: ComparisonofCumulativeResponseTimes inUnknownTerrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 A.40 Partial Zero-Swaps Evaluated Before Iteration 1 for Example in Figure A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 A.41 Partial One-Swaps Evaluated Before Iteration 2 for Example in Figure A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 A.42 Complex-Task Auctions: Comparison of Team Costs in Outdoor-Like Terrain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 329 A.43 Complex-Task Auctions: Comparison of Response Times in Outdoor- LikeTerrain(DisjointCoalitions). . . . . . . . . . . . . . . . . . . . . 329 A.44 Complex-Task Auctions: Comparison of Team Costs in Empty Terrain (DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 A.45 Complex-TaskAuctions: ComparisonofResponseTimesinEmptyTer- rain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . . . 330 A.46 Complex-Task Auctions: Comparison of Team Costs in Indoor-Like Terrain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 331 A.47 Complex-Task Auctions: Comparison of Response Times in Indoor- LikeTerrain(DisjointCoalitions). . . . . . . . . . . . . . . . . . . . . 331 A.48 Complex-TaskAuctions: ComparisonofTeamCostsforDifferentRobot Simple-TargetCapacities(DisjointCoalitions) . . . . . . . . . . . . . . 332 A.49 Complex-Task Auctions: Comparison of Response Times for Different RobotSimple-TargetCapacities(DisjointCoalitions) . . . . . . . . . . 332 A.50 Complex-TaskAuctions: ComparisonofTeamCostsforIdealReaction- FunctionCalculations(DisjointCoalitions) . . . . . . . . . . . . . . . 333 xvii A.51 Complex-TaskAuctions: ComparisonofResponseTimesforIdealReaction- FunctionCalculations(DisjointCoalitions) . . . . . . . . . . . . . . . 333 A.52 Complex-Task Auctions: Comparison of Team Costs for Loose Robot Simple-TargetCapacityConstraints(DisjointCoalitions) . . . . . . . . 334 A.53 Complex-Task Auctions: Comparison of Response Times for Loose RobotSimple-TargetCapacityConstraints(DisjointCoalitions) . . . . 334 A.54 Complex-Task Auctions: Comparison of Team Costs in Unknown Ter- rain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . . . 335 A.55 Complex-Task Auctions: Comparison of Numbers of Repeated Execu- tionsinUnknownTerrain(DisjointCoalitions) . . . . . . . . . . . . . 335 A.56 Complex-Task Auctions: Comparison of Cumulative Response Times inUnknownTerrain(DisjointCoalitions) . . . . . . . . . . . . . . . . 336 A.57 Complex-Task Auctions: Comparison of Team Costs in Outdoor-Like Terrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . 336 A.58 Complex-Task Auctions: Comparison of Response Times in Outdoor- LikeTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . 337 A.59 Complex-Task Auctions: Comparison of Team Costs in Empty Terrain (OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . . . . . 337 A.60 Complex-TaskAuctions: ComparisonofResponseTimesinEmptyTer- rain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 338 A.61 Complex-Task Auctions: Comparison of Team Costs in Indoor-Like Terrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . 338 A.62 Complex-Task Auctions: Comparison of Response Times in Indoor- LikeTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . 339 A.63 Complex-TaskAuctions: ComparisonofTeamCostsforDifferentRobot Simple-TargetCapacities(OverlappingCoalitions) . . . . . . . . . . . 339 A.64 Complex-Task Auctions: Comparison of Response Times for Different RobotSimple-TargetCapacities(OverlappingCoalitions) . . . . . . . . 340 A.65 Complex-TaskAuctions: ComparisonofTeamCostsforIdealReaction- FunctionCalculations(OverlappingCoalitions) . . . . . . . . . . . . . 340 xviii A.66 Complex-TaskAuctions: ComparisonofResponseTimesforIdealReaction- FunctionCalculations(OverlappingCoalitions) . . . . . . . . . . . . . 341 A.67 Complex-Task Auctions: Comparison of Team Costs for Loose Robot Simple-TargetCapacityConstraints(OverlappingCoalitions) . . . . . . 341 A.68 Complex-Task Auctions: Comparison of Response Times for Loose RobotSimple-TargetCapacityConstraints(OverlappingCoalitions) . . 342 A.69 Complex-TaskAuctions: ComparisonofTeamCostsforDifferentNum- bersofComplexTargets(OverlappingCoalitions) . . . . . . . . . . . 342 A.70 Complex-Task Auctions: Comparison of Response Times for Different NumbersofComplexTargets(OverlappingCoalitions) . . . . . . . . . 343 A.71 Complex-Task Auctions: Comparison of Team Costs in Unknown Ter- rain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 343 A.72 Complex-Task Auctions: Comparison of Numbers of Repeated Execu- tionsinUnknownTerrain(OverlappingCoalitions) . . . . . . . . . . . 344 A.73 Complex-Task Auctions: Comparison of Cumulative Response Times inUnknownTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . 344 A.74 Comparison of Team Costs of SARF (Simple-First and Complex-First) AlgorithmswithSKAuctionsforMiniMaxTeamObjective . . . . . . 345 A.75 ComparisonofSKAuctionsandSARF(Mixed)AlgorithmforExample inFigureA.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 A.76 Comparison of Team Costs of SARF (Complex-First) Algorithm with SKAuctions(Complex-First)forMiniLatTeamObjective . . . . . . . 349 A.77 Number of Simple Targets Allocated to Robots with Robot Simple- TargetCapacities∞(DisjointCoalitions) . . . . . . . . . . . . . . . . 355 A.78 Number of Simple Targets Allocated to Robots with Robot Simple- TargetCapacities∞(OverlappingCoalitions) . . . . . . . . . . . . . . 365 A.79 NumberofRelatedComplexTargets . . . . . . . . . . . . . . . . . . . 366 A.80 Complex-Task Negotiations: Comparison of Team Costs in Outdoor- LikeTerrain(DisjointCoalitions). . . . . . . . . . . . . . . . . . . . . 377 A.81 Complex-TaskNegotiations: ComparisonofResponseTimesinOutdoor- LikeTerrain(DisjointCoalitions). . . . . . . . . . . . . . . . . . . . . 377 xix A.82 Complex-TaskNegotiations: ComparisonofTeamCostsinEmptyTer- rain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . . . 378 A.83 Complex-TaskNegotiations: Comparison of Response Timesin Empty Terrain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 378 A.84 Complex-TaskNegotiations: ComparisonofTeamCostsinIndoor-Like Terrain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 379 A.85 Complex-TaskNegotiations: ComparisonofResponseTimesinIndoor- LikeTerrain(DisjointCoalitions). . . . . . . . . . . . . . . . . . . . . 379 A.86 Complex-Task Negotiations: Comparison of Team Costs for Different RobotSimple-TargetCapacities(DisjointCoalitions) . . . . . . . . . . 380 A.87 Complex-Task Negotiations: Comparison of Response Times for Dif- ferentRobotSimple-TargetCapacities(DisjointCoalitions) . . . . . . . 380 A.88 Complex-TaskNegotiations: ComparisonofTeamCostsforIdealReaction- FunctionCalculations(DisjointCoalitions) . . . . . . . . . . . . . . . 381 A.89 Complex-Task Negotiations: Comparison of Response Times for Ideal Reaction-FunctionCalculations(DisjointCoalitions) . . . . . . . . . . 381 A.90 Complex-TaskNegotiations: ComparisonofTeamCostsforLooseRobot Simple-TargetCapacityConstraints(DisjointCoalitions) . . . . . . . . 382 A.91 Complex-TaskNegotiations: ComparisonofResponseTimesforLoose RobotSimple-TargetCapacityConstraints(DisjointCoalitions) . . . . 382 A.92 Complex-Task Negotiations: Comparison of Team Costs for Random- izedInitialAllocations(DisjointCoalitions) . . . . . . . . . . . . . . . 383 A.93 Complex-Task Negotiations: Comparison of Response Times for Ran- domizedInitialAllocations(DisjointCoalitions). . . . . . . . . . . . . 383 A.94 Complex-Task Negotiations: Comparison of Team Costs in Unknown Terrain(DisjointCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 384 A.95 Complex-TaskNegotiations: ComparisonofNumbersofRepeatedExe- cutionsinUnknownTerrain(DisjointCoalitions) . . . . . . . . . . . . 384 A.96 Complex-TaskNegotiations: ComparisonofCumulativeResponseTimes inUnknownTerrain(DisjointCoalitions) . . . . . . . . . . . . . . . . 385 xx A.97 Complex-Task Negotiations: Comparison of Team Costs in Outdoor- LikeTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . 385 A.98 Complex-TaskNegotiations: ComparisonofResponseTimesinOutdoor- LikeTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . 386 A.99 Complex-TaskNegotiations: ComparisonofTeamCostsinEmptyTer- rain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . . . 386 A.100Complex-TaskNegotiations: Comparison of Response Timesin Empty Terrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . 387 A.101Complex-TaskNegotiations: ComparisonofTeamCostsinIndoor-Like Terrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . 387 A.102Complex-TaskNegotiations: ComparisonofResponseTimesinIndoor- LikeTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . 388 A.103Complex-Task Negotiations: Comparison of Team Costs for Different RobotSimple-TargetCapacities(OverlappingCoalitions) . . . . . . . . 388 A.104Complex-Task Negotiations: Comparison of Response Times for Dif- ferentRobotSimple-TargetCapacities(OverlappingCoalitions) . . . . 389 A.105Complex-TaskNegotiations: ComparisonofTeamCostsforIdealReaction- FunctionCalculations(OverlappingCoalitions) . . . . . . . . . . . . . 389 A.106Complex-Task Negotiations: Comparison of Response Times for Ideal Reaction-FunctionCalculations(OverlappingCoalitions) . . . . . . . 390 A.107Complex-TaskNegotiations: ComparisonofTeamCostsforLooseRobot Simple-TargetCapacityConstraints(OverlappingCoalitions) . . . . . . 390 A.108Complex-TaskNegotiations: ComparisonofResponseTimesforLoose RobotSimple-TargetCapacityConstraints(OverlappingCoalitions) . . 391 A.109Complex-Task Negotiations: Comparison of Team Costs for Different NumberofComplexTargets(OverlappingCoalitions) . . . . . . . . . 391 A.110Complex-Task Negotiations: Comparison of Response Times for Dif- ferentNumberofComplexTargets(OverlappingCoalitions) . . . . . . 392 A.111Complex-Task Negotiations: Comparison of Team Costs for Random- izedInitialAllocations(OverlappingCoalitions) . . . . . . . . . . . . . 392 xxi A.112Complex-Task Negotiations: Comparison of Response Times for Ran- domizedInitialAllocations(OverlappingCoalitions) . . . . . . . . . . 393 A.113Complex-Task Negotiations: Comparison of Team Costs in Unknown Terrain(OverlappingCoalitions) . . . . . . . . . . . . . . . . . . . . . 393 A.114Complex-TaskNegotiations: ComparisonofNumbersofRepeatedExe- cutionsinUnknownTerrain(OverlappingCoalitions) . . . . . . . . . . 394 A.115Complex-TaskNegotiations: ComparisonofCumulativeResponseTimes inUnknownTerrain(OverlappingCoalitions) . . . . . . . . . . . . . . 394 A.116Comparison of Numbers of Profitable Task Exchanges and Negotiation RoundsforInitialAllocationsProducedbySARF(Mixed)Algorithm . 395 A.117TeamCostsandResponseTimesforSNRFAlgorithmswithDiscretiza- tionGranularity10(DisjointCoalitions) . . . . . . . . . . . . . . . . . 400 A.118Number of Simple Targets Allocated to Robots with Robot Simple- TargetCapacities∞(DisjointCoalitions) . . . . . . . . . . . . . . . . 403 A.119Comparison of Numbers of Profitable Task Exchanges and Negotiation RoundsforInitialAllocationsProducedbyRandomizedAlgorithm . . 407 A.120Number of Simple Targets Allocated to Robots with Robot Simple- TargetCapacities∞(OverlappingCoalitions) . . . . . . . . . . . . . . 416 A.121ComparisonofDifferentDiscretizationGranularities(OverlappingCoali- tions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 A.122Response Times of SNRF Algorithms with 7 Complex Targets (Over- lappingCoalitions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 xxii Chapter1 Introduction Over the past decade, a significant research effort has been devoted to the field of multi-robot systems. In the near future, hopefully, a team of cooperative robots will be able to explore the surface of Mars, deliver parcels to customers’ homes, and perform search-and-rescue operations after disasters. However, it is non-trivial to coordinate a team of robots well in a distributed computing environment especially when there are positive and negative synergies 1 among tasks and some tasks need to be executed by more than one robot simultaneously. In this dissertation, we study how to allocate and re-allocate such tasks quickly to distributed robots based on their local computations and interactions so that the overall team performance is as high as possible. The con- tract net protocol [Smi79] is a classic interaction paradigm for coordinating distributed robotsandhasrecentlyevolvedtoseveralmoresophisticatedcoordinationmechanisms for decentralized task allocation, such as auction-like and negotiation-like protocols [TLJK05,San93]. Thisdissertationadvancesthestateoftheartinthisresearchareaby developingseveralnoveltask-allocationalgorithmsthatgreedilyallocateandre-allocate taskstorobotsbasedonauction-likeandnegotiation-likeprotocols. 1.1 Task-AllocationProblem Given a team of distributed robots, a set of tasks, and some definition of team perfor- mance, solving the task-allocation problem is to answer the question: ”Which robots 1 Twotasksaresaidtoexhibitapositive(negative)synergyforarobotifthecostoftherobottoexecute bothtaskstogetherissmaller(larger)thanthesumofitscostsofexecutingthemindividually. 1 should execute which tasks and when should they execute the tasks so that the team performanceisashighaspossible?” In this dissertation we study two types of tasks that need to be executed by robots, namely,simpletasksandcomplextasks: • A simple task is one that can be executed by single robots. A robot can freely determinewhentoexecuteitsallocatedsimpletasks. • Acomplextaskisonethathastobeexecutedbymorethanonerobotsimultane- ously. A complex task is always allocated to a group of robots, and the robots in that group have to coordinate with each other to agree on the common execution time. If a task-allocation problem consists of only simple tasks, then we call it a simple- task allocation problem. Otherwise, we call it a complex-task allocation problem. In thisdissertation,westudybothtypesoftask-allocationproblems. When solving a task-allocation problem, we are interested in minimizing a metric, calledtheteamcostoftherobotteam,whosedefinitiondependsonauser-definedteam objective that characterizes the team performance of the robot team. In this disserta- tion, we consider the following three team objectives that are commonly studied in the literature: • MiniSum Team Objective [DS02, GMR97a, SB07, PKM + 03]: In a task- allocation problem, each robot incurs some domain-dependent cost for executing its allocated tasks. The MiniSum team objective is to minimize the sum of the costsofallrobotsforexecutingtheirallocatedtasks. • MiniMaxTeamObjective[TRS04,EGK + 04,Sur10]: TheMiniMaxteamobjec- tiveistominimizethelargestcostofanyrobotforexecutingitsallocatedtasks. 2 • MiniLat Team Objective [BCC + 94, AABV99, FHR03]: In a task-allocation problem, the latency of any task is defined to be the time when it is executed by robots, namely, the execution time of the task. The MiniLat team objective is tominimizethesumofthelatenciesofalltasks. 1.1.1 MotivatingDomain: Multi-RobotRouting Now, we introduce a key domain that motivates our work, namely, multi-robot rout- ing. A multi-robot routing problem consists of a number of robots (that represent vehi- cles, mobile agents, or other objects that can move in a given either initially known or unknown terrain) and a number of targets at different locations of the terrain (that need tobevisitedbyrobots). Therearetwotypesoftargetsinamulti-robotroutingproblem, whichcorrespondtothetwotypesoftasksinatask-allocationproblem. Asimpletarget needs to be visited by a single robot, and a complex target needs to be visited by more thanonerobotsimultaneously. The left part of Figure 1.1 presents a small example of multi-robot routing in lunar base construction scenarios [THWMA05, DSLF05, BBC02]. In this example, three lunar rovers (robots) located at two space stations need to visit different locations to performvariousconstructionoperations(targets). Therearetwotypesofoperations. A single lunar rover can handle a simple operation (simple target), such as mining open pits, deploying pipes, and crushing stones by itself. However, at least two lunar rovers need to work together to perform a complex operation (complex target), such as con- structing space tents [BBC02]. A solution of this problem is to allocate these targets to roverssothatalloperationscanbesuccessfullyperformed. TherightpartofFigure1.1 presents this problem on a four-neighbor planar grid with square cells and one solution fortheallocationsoftargetstorobots(withtheirresultingroutes). 3 Mining open pits Constructing space tent Lunar rover Lunar rover Crushing stones Crushing stones Deploying pipes Lunar rover Simple target Robot Robot Robot Simple target Simple target Simple target Complex target Figure1.1: Example1ofMulti-RobotRouting Multi-robot routing is a realistic problem [KS08, FMPS95, FMP96, VKP10] since onehastosolveinstancesofmulti-robotroutinginmanyreal-worldapplications: • PlanetaryRoverCoordination[Ker90,TONG + 02]: AteamofMarsrovershas toperformon-sitegeologicalinvestigationsonthesurfaceofMars. Forexample, theymayberequiredtotakerockprobesatseveraldifferentsites. Ifwewantthe totalenergyusedbyMarsroversfortravelingorwaitingtobeassmallaspossible, thentheroverteamisfacingamulti-robotroutingproblemwiththeteamobjective tominimizethetotalenergyusedbyrobots. • Fire Fighting Disaster Response [JDS07, RHL10]: A team of fire engines has toparticipateinthedisasterpreparednessandemergencyresponseforunexpected natural disasters or terrorist attacks. For example, they may be required to extin- guish fires at several locations. If we want fire engines to extinguish all fires as quicklyaspossible,thenthefireengineteamisfacingamulti-robotroutingprob- lemwiththeteamobjectivetominimizethetask-completiontime. • SearchandRescue[KTN + 99,RFMJ10,NSK + 05]: Ateamofrescuerobotshas tosearchforvictimsafteranearthquakehappensinacity. Forexample,theymay be required to search a large number of collapsed buildings to locate survivors. If we want buildings to be visited and searched by the rescue team as early as 4 possible, then the rescue team is facing a multi-robot routing problem with the teamobjectivetominimizethesumofthevisittimesofallbuildings. Inamulti-robotroutingproblem,wedefinethepathcostofarobottobethesmallest timetherobotneedstovisititsallocatedtargets,includingboththetraveltimebetween targetsandthewaittimeforotherrobotsatcomplextargets. 2 Wedefinethelatencyofa targettobethevisittimewhentherequirednumberofrobot(s)havearrivedatit. Then, thethreeteamobjectivesinmulti-robotroutingcanbeexplainedasfollows: • The MiniSum team objective is to minimize the team cost which is defined to be the sum of the path costs of all robots for visiting all targets. The measure of this team objective corresponds to the total energies used by robots in our Mars rover example. • The MiniMax team objective is to minimize the team cost which is defined to be thelargestpathcostofanyrobotforvisitingalltargets. Themeasureofthisteam objectivecorrespondstothetask-completiontimeinourfirefightingexample. • TheMiniLatteamobjectiveistominimizetheteamcostwhichisdefinedtobethe sumofthelatenciesofalltargets. Themeasureofthisteamobjectivecorresponds tothesumofthevisittimesofallbuildingsinoursearchandrescueexample. Generallyspeaking,onecannothopetoalwaysachieveallocationswiththesmallest teamcostforatask-allocationproblemwithsynergiesbetweentasks. Forexample,itis knowntobeNP-hardtosolvemulti-robotroutingproblemsforallthreeteamobjectives [LMK + 05, Sur10, Sit02]. Since it is impractical to pursue the smallest team cost in the real world, in this dissertation we develop heuristic task-allocation algorithms that can 2 Weassumethatrobotscanleaveatargetimmediatelyaftertherequirednumberofrobotsvisitingthe targettogether. (Wecansimplymodifythetraveltimebetweentargetstoincorporatethetimerobotsneed tostayattargetsifthisassumptiondoesnotholdforanymulti-robotroutingproblem.) 5 Mi ss i o n Plan n in g Ta s k Al lo cat io n E xec ut io n t as k s , ag e nt s , c o n s train ts allo c a t io ns s c he d u l es ne w kn ow l ed g e Lo cal Sc h e dul in g Figure1.2: AGeneralModelofDistributedProblemSolving trade-off the response time (the time the algorithm needs to solve the task-allocation problem) and the team cost (the quality of the resulting allocations). Thus, the goal of this dissertation is to develop task-allocation algorithms that are able to produce good allocations(theteamcostissmall)withinreasonableresponsetimes(inseconds). Given that it is easy to understand a multi-robot routing problem and its solutions, we will use examples of multi-robot routing to illustrate our task-allocation algorithms developedinthisdissertation. Also,giventhatitiseasytogeneratemulti-robotrouting problemsbychangingthelocationsofrobotsandtargets,wewillusemulti-robotrouting asthetestdomaintoevaluateourtask-allocationalgorithms. 1.1.2 TaskAllocationinDistributedProblemSolving Howtoallocatetaskstorobotswellisacentralresearchchallengeindistributedproblem solving [DR94, Dur99, Les02], which studies how to coordinate multiple cooperative robots to accomplish a team mission. It is widely believed that a team of robots is capable of executing tasks better than a single robot since multiple robots can execute tasks in parallel and can execute tasks that single robots cannot [LMK + 05, DZZS04]. However,itisnon-trivialtocoordinatemultiplerobotswellinadistributedfashioneven 6 whenallrobotsarecooperative[BMBB09]. Figure1.2showsageneraltop-downmodel [Zlo06,DS83]ofdistributedproblemsolving. Itcontainsthefollowingfourphases: • Mission Planning Given a team mission that requires the cooperation of multi- ple robots, a high-level planner has to analyze the mission, break it down into multiple small tasks, and specify the constraints of these tasks, for example that a complex task cannot be decomposed any further and needs to be executed by a given number of robots at the same time. The planner may also have to select a teamofsuitablerobotsthatarequalifiedforexecutingthesetasks. Thisphasecor- responds to the ”task division” stage in [DS83] and is similar to the combination ofthe”teamdesign”and”missionplanning”proceduresin[Zlo06]. • TaskAllocationAfterthepreviousphase,wearegivenateamofrobotsandaset of tasks with constraints. Then, the algorithms in the task-allocation phase need toallocatetaskstorobotssothatalltaskscanbeexecutedbyrobotsasquicklyas possibleandtheresultingteamcostisassmallaspossible. • LocalScheduling After all tasks have been allocated to robots, each robot needs to determine a schedule to execute its allocated tasks subject to their constraints. Although this phase is called local scheduling, robots may still have to interact witheachother. Forexample,agroupofrobotshastocoordinatewitheachother toagreeontheexecutiontimeofacomplextaskthatisallocatedtothem. • ExecutionWhenrobotshavetheirschedulesready,theystarttoexecutetheirallo- cated tasks. Whenever new knowledge becomes available during the execution phase, for example, a robot detects that its route is blocked, the task-allocation phase is re-entered to allocate the remaining tasks (that have not yet been exe- cuted)torobots. 7 Mi ss i o n Plan n in g Task Al lo c ation E xec ut io n t as k s , ag e nt s , c o n s train ts allo c a t io ns s c he d u l es ne w kn ow l ed g e Lo cal Sc h e dul in g All o c a ti n g T a s ks R e-Alloc a tin g T asks ini tial a l lo cati on s Figure1.3: TwoStagesofTask-AllocationPhase In the literature, the task-allocation phase is further decomposed into the following twosequentialstages[Ger03,Sar07,Dia04,Zlo06],asshowninFigure1.3. • AllocatingTasksAlltasksthathavenotbeenexecutedbyrobotsareunallocated at the beginning of this stage. Then, they are allocated to robots and do not get re-allocatedinthisstage. • Re-AllocatingTasksAlltasksthathavenotbeenexecutedbyrobotsshouldhave already been allocated at the beginning of this stage. We call such allocations to be initial allocations. Then, the initial allocations might be further optimized by re-allocation techniques in this stage, for example, performing multiple task exchangesamongmultiplerobots. These two stages are complementary: On one hand, if the algorithms for allocating tasks work well, namely, the team cost of the initial allocations after the first stage is already smallest or very close to smallest, then the second stage of re-allocating tasks can be optional. On the other hand, if we have effective algorithms for re-allocating tasksinthesecondstage,thentheteamcostofthefinalallocationsafterthesecondstage shouldstillbesmallestorveryclosetosmallestnomatterhowtheinitialallocationsare 8 produced (such as random allocations). In this dissertation, we study heuristic task- allocationalgorithmsinbothstagesofthetask-allocationphase. 1.1.3 EvaluationCriteria Wewantthetask-allocationalgorithmsdevelopedinthisdissertationtobebothefficient and effective. We have the following criteria to empirically evaluate a task-allocation algorithm: • Response Time We want the algorithm to allocate tasks to robots as quickly as possible. The response times of real-world applications should be realistic (in seconds) since robots should not be idle for a long time if they are working on a time-criticalmissionsuchassearchandrescue. • TeamCostWewanttheteamcostoftheresultingallocationsofthealgorithmto beassmallaspossible. 1.2 SequentialSingle-ItemAuctions Sequential single-item auctions have been studied by several research groups for solv- ing multi-robot task-allocation problems. Sequential single-item auctions take several rounds to allocate tasks to robots. All tasks are initially un-allocated. In each round, every robot bids on at most one un-allocated task and the auctioneer allocates one task tosomerobotsothatthecurrentteamcostincreasestheleast. Researchers have already provided several theoretical results on sequential single- item auctions. For example, for solving the multi-robot routing problems in known ter- rainwiththeMiniSumteamobjective, theresultingteamcostofsequentialsingle-item auctionsisguaranteedtobeonlyaconstantfactorawayfromthesmallest[LBK + 04]. To 9 ourknowledge,thesearethefirstanalyticalresultsabouttheteamperformanceresulting fromusingauctionalgorithmsforrobotcoordination. Sequential single-item auctions are also simple to understand and easy to imple- ment. Many research groups have implemented and evaluated sequential single-item auctions for robot coordination in their multi-robot systems. For example, Iocchi et al. [IFN14]implementedsequentialsingle-itemauctionsforsolvingmulti-robotpatrolling problems. Gonzalez et al. [GCM14] implemented sequential single-item auctions for coordinating multiple Unmanned Aerial Vehicles (UAVs). Schneider et al. [SBO + 14] implemented sequential single-item auctions for coordinating physical robots and eval- uated their performance in office-like environments. Schoenig and Pagnucco [SP11] implemented sequential single-item auctions in dynamic environments where tasks are generated over time. Boutilier et al. [BGMS10] implemented sequential single-item auctions for allocating resources to robots. Guerrero and Oliver [GO11] implemented sequential single-item auctions for solving multi-robot foraging problems. Mosteo and Montano [MM07] performed comparative experiments of sequential single-item auc- tionsfortheMiniMaxandMiniSumteamobjectives. Many research groups also extend sequential single-item auctions to increase their capabilities. For example, Mosteo et al. [MML09] extended sequential single-item auctions to robots that have limited communication ranges. Kishimoto and Sturtevant [KS08]studiedandcomparedseveraldifferentpathplanningtechniqueswhenapplying sequential single-item auctions to multi-robot routing. Koenig et al. [KZT + 08] studied and compared the regret clearing strategy used by the auctioneer in the winner deter- minationphasewiththeoriginalgreedyclearingstrategyusedinsequentialsingle-item auction. Heap and Pagnucco [HP12] studied and compared the cluster bidding strat- egy used by robots in the bid construction phase with the original single-task bidding strategyusedinsequentialsingle-itemauctions. 10 Economists have studied auctions for many decades in order to efficiently allo- cate scarce items (resources or goods) among rational agents (individuals and groups) [Mil89]. In their settings, they usually assume that agents are self-interested, competi- tive,anddonotwanttorevealtheirprivateevaluations. Asaconsequence,theresearch efforts in economics mainly focus on strategic bidding rules (in order to maximize the utilities of individual agents) and mechanism design (in order to provide agents incen- tives to truthfully reveal their private information). In multi-robot systems, robots are usuallydesignedtobecooperativetoachieveacommonteammission. Therefore,robots insequentialsingle-itemauctionsandothertask-allocationalgorithmsdiscussedinthis dissertationcanbetrustedtoprovidetheirrealcostsandexecutetheirallocatedtasks. Theideaofusingauctionalgorithmstocoordinatemultiplerobotshasexistedsince the development of the Contract Net Protocol around 1980 [Smi79]. Since robots are cooperative and always collaborate to minimize the team cost, the term ”auction” in these algorithms is mostly used to describe the interaction protocol among robots: i) robots in the bidding phase evaluate and submit bids to the auctioneer; and ii) the auc- tioneer in the winner-determination phase makes allocation decisions. Auction algo- rithmsbecomepromisinginroboticsbecausethattheyprovideatrade-offbetweencen- tralized and distributed approaches: the majority of the computation efforts are split among the robots in the bidding phase and the centralized auctioneer makes unam- biguous and consistent allocation decisions for the whole robot system in the winner- determination phase. (See more discussions on the advantages of auction algorithms in [GM02, DZKS06].) From our perspective, auction algorithms provide one alterna- tive approach of solving multi-robot task-allocation problems. Zlot [Zlo06] identified several requirements for a robot system to qualify for auction algorithms. However, it is still an open problem when such auction algorithms are more feasible and preferred 11 than other approaches. In this dissertation, we decided on extending sequential single- itemauctionsnotonlybecausetheyaresimpletounderstandandimplementortheyare popularinrobotics,butalsobecausetheydemonstrateseveralinspiringproperties: • Sequential single-item auctions make the same allocation decisions as the simple centralized greedy approach. In order to solve multi-robot task-allocation prob- lems, one can naturally come up with a greedy approach in which a centralized planner in each round evaluates all possible allocations of single tasks to robots and then chooses the one that increases the current team cost the least. Sequen- tial single-item auctions achieve the same allocation decisions of this centralized greedyapproachbydecomposingthedecisionprocessintotwophasesoftheauc- tionprotocol: robotslocallyevaluateandsubmittheirsmallestbidsonsingletasks to the auctioneer and then the auctioneer chooses the overall smallest bid as the winner. • Robotsinsequentialsingle-itemauctionsevaluateallun-allocatetasksbutsubmit onlyonebidtotheauctioneerineachround. Sincetheauctioneerwilleventually choose the overall smallest bid among all possible allocations of single tasks to robotsasthewinnerineachround,robotscanlocallyinferthattheirnon-smallest bids on un-allocated tasks have no chance to win. As a consequence, robots sub- mitonlytheirsmallestbids. Inotherwords,robotscommunicateonlyasmallpart oftheirlocalcomputationresultstotheauctioneer. • The runtime of the auctioneer in sequential single-item auctions to determine the winningbidineachroundislinearinthenumberofrobotsandindependentofthe number of the un-allocated tasks. Since robots in sequential single-item auctions submit at most one bid (i.e. their smallest bid) to the auctioneer, the number of 12 bids the auctioneer needs to evaluate in each round is bounded by the number of robots. Itisverynaturalforustoaskwhetherwecankeepthesepropertieswhensequential single-item auctions are extended. In this dissertation, we choose to extend sequential single-itemauctionsinthefollowingtwodirections: • Weextendsequentialsingle-itemauctionssothattheyarelessgreedyinallocating simple tasks to robots. In the other words, we propose sequential single-item auctions be able to allocate upto k additional (instead of single) tasks to robots so that the team cost increases the least in each round. By increasing the number of tasks allocated to robots in each round, we expect that more synergies among taskscanbeevaluatedandexpressedinthebidssubmittedbyrobotsandthusthe auctioneer can find better allocations compared to the original sequential single- itemauctions. Onenaiveapproachissimplymakingeachrobotevaluateandalso bid on each bundle of at mostk un-allocated tasks to the auctioneer, and then let theauctioneerevaluateallpossibleallocationsofuptoktaskstorobotsandfinally choose the one that increases the current team cost the least. In this dissertation, wewanttoinvestigatewhetherrobotsneedtosubmitalltheircomputationresults totheauctioneerinordertomakesuchallocationdecisions. • Weextendsequentialsingle-itemauctionssothattheycanallocatecomplextasks thatneedtobeexecutedbyseveralrobotssimultaneously. Intheotherwords,we propose sequential single-item auctions be able to allocate one complex (instead ofsimple)tasktorobotssothattheteamcostincreasestheleastineachround. In terms of Gerkey and Mataric’s taxonomy [GM04] on multi-robot task-allocation problems, we extend sequential single-item auctions from allocating single-robot 13 (SR) tasks to multi-robot (MR) tasks. There are existing applications in the liter- aturethathavetoallocatesuchcomplexmulti-robottaskstorobots. Forexample, Donaldetal. [DJR97]andGerkey[Ger03]studiedcooperativeboxpushingtasks whereasinglerobotisnotstrongenoughtomovetheboxitself. ParkerandTang [PT06] studied multi-robot tasks where heterogeneous robots need to share their sensors. Fua and Ge [FG05] studied reconnaissance tasks where robots have to cooperativelyclearanobstructingrockdetectedduringthemission. Knepperetal. [KLRR13]studiedfurnitureassemblytaskswheremultiplerobotshavetocollab- oratetoassembleantabletogether. Therearealsootherpotentialapplicationsthat may require coalitions of robots to execute tasks simultaneously. For example, several pursuers in pursuit-evasion problems may attempt to capture the intruder fromdifferentdirections[HWW + 09]. Severalrescuerobotsinsearch-and-rescue problems may work together to save survivors trapped in accidents or disasters [JWE97]. Several military de-mining robots may gather together to detect and disarm mines in the field [Kim11]. Several lunar robots may collaborate with each other for performing lunar base construction tasks [SHO + 05, BBC02]. One naiveapproachissimplymakingeachrobotevaluateandalsobidoneachpossible execution time of the complex task to the auctioneer, and then let the auctioneer evaluate all possible allocations of the complex task to robots at all possible exe- cution times and finally choose the one that increases the current team cost the least. Inthisdissertation,wewanttoinvestigatewhetherrobotsneedtosubmitall their computation results to the auctioneer in order to make the same allocation decisions. 14 T ask Pla n n in g I n itiator P ar ti c ip ant Lo cal E v al uat io n c a l l f o r p r o p o s a l s Pr o po sa l Ev a lua ti o n r e s p o n s e ( r e f u s e / p r o p o s e ) n o t i f i c a t i o n ( r e j e c t / a c c e p t ) Sch e du le U pda te Figure1.4: InteractionDiagramofContractNetProtocol 1.3 OurApproach In this section, we describe our approach used in this dissertation for developing task- allocation algorithms. We first describe the system framework of our task-allocation algorithms, namely, how robots interact with each other in a task-allocation algorithm. After that, we discuss the performance principle of our task-allocation algorithms, namely,whichtasksshouldbeallocatedtowhichrobotsbyatask-allocationalgorithm. 1.3.1 SystemFrameworks In this dissertation, we choose the contract net protocol [Smi79, DS83, SD81] as the systemframeworkofourtask-allocationalgorithms. Thecontractnetprotocolisoneof the most important paradigms developed in distributed problem solving for the decen- tralizedcontrolofmultiplerobotsandprovidesaformalprocedurefordevelopingcoor- dination mechanisms in a multi-robot system [SD78, SD80, Dyk87, Bak88, SKB97, OHB + 98, SS04, CBER00, CD05, CDGP05, FKMP95, YHH + 98, HPX07]. Figure 1.4 showsthehigh-levelinteractiondiagrambetweenrobotsintheoriginalcontractnetpro- tocol. There are two types of roles for robots, namely, an initiator and a participant. A robot can be either an initiator or a participant. Figure 1.4 illustrates the interaction sequence between an initiator and a participant. The contract net protocol consists of thefollowingfourphases: 15 • Local Planning [Initiator] The initiator first analyzes its current workload with its capacities, capabilities and other constraints. If the initiator finds out that it cannot execute a particular task or it may be more beneficial to sub-contract the task to other robots, then it broadcasts a ”call for proposals” message to all other robots(participants). The”callforproposals”messagecontainsthedescriptionof the task and the specification for expected proposals. After sending the ”call for proposals”message,theinitiatorwaitsforproposalsandthecontrolofthesystem istransferredtotheparticipantswhoreceivedthismessage. • Local Evaluation [Participant] Each participant evaluates the task specified in the”callforproposals”message. Basedontheevaluationoutcome,theparticipant may either refuse the initiator indicating that it cannot execute the task or submit aproposaltotheinitiatorthatindicatesitsappropriatenessforexecutingthetask. Aftersendingtheproposalmessage,theparticipantwaitsforthenotificationfrom theinitiatorandthecontrolofthesystemistransferredbacktotheinitiator. • Proposal Evaluation [Initiator] The initiator evaluates all proposals submitted by participants. 3 Based on the evaluation outcome, the initiator determines the participant that is most appropriate for executing the task and sends an accept notification to it. All other participants receive a reject notification. After this phase, the task is allocated to the ”winning” participant and the initiator has no responsibilityforthetaskanylonger. 4 3 If no proposal is submitted, which indicates that no participants can execute the task in the ”call for proposals” message, then the initiator has to either execute the task by itself or report ”failure” to the high-levelmissionplannerifitcannotexecutethetask. 4 In the complete description of the contract net protocol [Smi79], the initiator still has to monitor the execution of the task, even after accepting a proposal, to ensure that the task is executed successfully. In thisdissertationweignorethisstepsinceitisbeyondthescopeofthetask-allocationphase. 16 Auctioneer Task Announcement Bidder Bid Construction list of unallocated tasks bid submission Winner Determination task awarding Schedule Update unallocated tasks exist Figure1.5: InteractionDiagramofIterativeAuctionProtocol r 1 x 1 x 2 r 2 Figure1.6: Example2ofMulti-RobotRouting • ScheduleUpdate[Participant]Ifaparticipantreceivesanacceptnotification,it updatesitsallocatedtasklistbyinsertingthenewlyallocatedtask. Recently,theoriginalcontractnetprotocolhasbeenextendedtoseveralauction-like and negotiation-like protocols [TLJK05, San93] for allocating and re-allocating tasks. Now,weformalizetheseprotocolsasfollows: Auctioneer Ta s k An n oun c em e n t Robot r 2 B id Co n st r uc t ion { x 1 , x 2 } { x 1 : 4 , x 2 : 2 } W i n n er Deter mi n at io n ( x 2 → r 2 ) Sch e d ule U pda te Robot r 1 { x 1 , x 2 } B id Co n s t r uction { x 1 : 3 , x 2 : 5 } Ta s k An n oun c em e n t ( x2 → r 2 ) Sch e d ule U pd at e B id Co n s t r uction B id Co n st r uc t ion { x 1 } { x 1 } { x 1 : 3 } W i n n er Deter mi n at io n { x 1 : 6 } Sch e d ule U pd at e Sc h edule U p dat e ( x 1 → r 1 ) ( x 1 → r 1 ) x 1 is s till u n allo c ate d Figure1.7: ExampleofIterativeAuctionProtocol 17 • Iterative Auction Protocol: As shown in Figure 1.5, the iterative auction proto- colspecifiesinteractionsbetweenanauctioneerandabidder,wheretheauctioneer allocatesmultipletaskstobiddersinoneormultiplerounds. Ithasasimilarinter- action diagram as the original contract net protocol, and its interaction sequence betweenanauctioneerandabidderalsoconsistsoffourphases: – Task Announcement [Auctioneer] The auctioneer generates the list of unallocatedtasksandthenannouncesittoallotherrobots(bidders). – Bid Construction [Bidder] Each bidder evaluates the unallocated tasks announcedbytheauctioneerandsubmitsbidstotheauctioneer. – Winner Determination [Auctioneer] The auctioneer evaluates the bids submitted by bidders and then allocates one or multiple unallocated tasks to the winning bidder(s). If there are any remaining unallocated tasks after thisphase,theauctioneerstartsanewroundoftheauction. – Schedule Update [Bidder] Each bidder updates its allocated task list by insertingthenewlyallocatedtasks(ifany). Figure 1.6 presents an example of multi-robot routing, which consists of two robots and two simple targets. Robots can move between any two unblocked (horizontally or vertically) adjacent square cells and such moves take unit time. Figure1.7illustratesonepossibleinteractionsequenceofrobotsiftheyfollowthe iterativeauctionprotocoltoallocatetargets. Theauctioneercanbeeitherrobotr 1 or robot r 2 or any other computing entity with which robots r 1 and r 2 can com- municate. 5 Inthisexample,theauctioneertakestworoundstoallocatetargetsx 1 5 How to select an aunctioneer is another important research topic [BA10, OSFM07] in multi-robot systems, which is beyond the scope of this dissertation. We assume that the auctioneer has already been selectedandisknowntoallrobotsatthebeginningofourtask-allocationalgorithms. 18 Initi ator T ask A n n o un ce m e n t Par t icipant Ta s k -Exch a n ge E v al ua tion l i s t o f a l l o c a t e d t a s k s p r o p o s a l s u b m i s s i o n Pr o p o sal E v al uat io n n o t i f i c a t i o n Sc h edule U p dat e Figure1.8: InteractionDiagramofOne-to-OneNegotiationProtocol andx 2 . Atthebeginningofeachround,theauctioneerannouncestheunallocated targetstobothrobots. Then,robotsbidtheirevaluationsonsingletargets. Finally, theauctioneerallocatesonetargettoonerobotineachround. • One-to-OneNegotiationProtocol: AsshowninFigure1.8,theone-to-onenego- tiationprotocolspecifiesinteractionsbetweenaninitiatorandaparticipant,where theyexchangetheirallocatedtasksineachround. Ithasasimilarinteractiondia- gram as the original contract net protocol, and its interaction sequence between aninitiatorandaparticipantalsoconsistsoffourphases: – Task Announcement [Initiator] The initiator generates the list of its allo- catedtasksandthenannouncesitwithotherlocalinformation(ifnecessary) toallotherrobots(participants). – Task-ExchangeEvaluation[Participant]Eachparticipantevaluatespoten- tialexchangesoftasksthatarecurrentlyallocatedtoitselfandthosethatare currently allocated to the initiator. After that, it proposes task exchanges withitsownevaluationstotheinitiator. – Proposal Evaluation [Initiator] The initiator evaluates all proposals sub- mittedbyparticipantsandacceptsatmostoneproposal. Afterthat,theinitia- tornotifieseachparticipantwhichproposalisaccepted. Finally,itupdatesits 19 Robot r 1 T ask A n n o un ce m e n t Robot r 2 Ta s k -Exch a n ge E v al ua tion { x 2 } { x1 ↔ x2 : 2 } Pr o p o sal E v al uat io n Sc h edule U p dat e { x 1 ↔ x 2 : a c c e p t } Figure1.9: ExampleofOne-to-OneNegotiationProtocol allocatedtasklistbyperformingthetaskexchangesspecifiedintheaccepted proposal(ifany). – ScheduleUpdate[Participant]Theparticipantwhoseproposalisaccepted updates its allocated task list by performing the task exchanges specified in theproposal. Consider again the multi-robot routing problem shown in Figure 1.6. Assume thattargetx 2 iscurrentlyallocatedtorobotr 1 andtargetx 1 iscurrentlyallocated to robot r 2 . Figure 1.9 illustrates one possible interaction sequence of robots if they follow the one-to-one negotiation protocol to re-allocate targets between them. Robotr 1 first announces its allocated targetx 2 to robotr 1 . Then, robotr 2 proposes the exchange of target x 2 and its allocated target x 1 with its evaluation value2torobotr 1 . Finally, robotr 1 acceptsthisproposalandnotifiesrobotr 2 to updateitsallocation. • Multi-Party Negotiation Protocol: As shown in Figure 1.10, the multi-party negotiation protocol specifies the interactions between an initiator and multiple participants. The multi-party negotiation protocol extends the one-to-one negoti- ationprotocolsothatrobotscanperformmultipletaskexchangesamongmultiple robotsineachround. Theinteractionsequenceofthemulti-partynegotiationpro- tocol between the initiator and n participants (n≥ 2) consists of the following phases: 20 Initiator Task Announcement Participant 1 Initial Task-Exchange Evaluation list of allocated tasks Proposal Evaluation Participant 2 Proposal Construction Participant n Proposal Construction list of allocated tasks list of allocated tasks proposal submission proposal submission . . . . . . Initial Task-Exchange Evaluation Initial Task-Exchange Evaluation proposal submission . . . notification notification notification . . . Schedule Update Schedule Update Schedule Update proposal submission Figure1.10: InteractionDiagramofMulti-PartyNegotiationProtocol – Task Announcement [Initiator] This phase is identical to the task- announcementphaseoftheone-to-onenegotiationprotocol. – Initial Task-Exchange Evaluation [Participant] Each participant evalu- atespotentialexchangesofthetasksthatarecurrentlyallocatedtoitselfand those that are not currently allocated to itself. After that, it proposes task exchangeswithitsownevaluationstootherparticipants. – Proposal Construction [Participant] Each participant considers each pro- posal submitted by others: The participant further constructs the proposal by updating and/or adding task exchanges involving its currently allocated tasks. After that, it updates the evaluation of the proposal and submits it 21 r 1 x 1 x 2 r 2 x 3 r 3 x 4 Figure1.11: Example3ofMulti-RobotRouting R ob ot r 2 Robot r 3 { x 2 } R ob ot r 1 { ( x 2 ↔ x 4 , x 3 ↔ x 1 ) : 7 } T ask An n o u n c emen t In it ia l Ta s k -Exch a n ge E v al ua tion Pr o pos a l Amen da ti o n Pr o po sal Ev a lua ti o n { ( x2 ↔ x 4 , x 3 ↔ x 1 ) : 7 } Sc h edule U p dat e Sc h edule U p dat e {(x 2 ↔ x 4 , x 3 ↔ x 1 ) : acc ept} {(x 2 ↔ x 4 , x 3 ↔ x 1 ) : a c ce p t} Figure1.12: ExampleofMulti-PartyNegotiationProtocol either to the initiator (if every task exchange in the proposal has been eval- uated by both robots involved in the task exchange), or to other participants (otherwise). – Proposal Evaluation [Initiator] The phase is identical to the proposal- evaluationphaseoftheone-to-onenegotiationprotocol. – Schedule Update [Participant] Each participant that is involved in the accepted proposal updates its allocated task list by performing the task exchangesspecifiedintheproposal. Figure 1.11 presents another example of multi-robot routing, which consists of three robots and four simple targets. Assume that targetx 2 is currently allocated 22 Stage1: AllocatingTasks(AuctionAlgorithms) Stage2: Re-AllocatingTasks(NegotiationAlgorithms) One-to-One/Multi-Party Simple-TaskAllocation IterativeAuctionProtocol NegotiationProtocol (Simple-TaskAuctions) (Simple-TaskNegotiations) One-to-One/Multi-Party Complex-TaskAllocation IterativeAuctionProtocol NegotiationProtocol (Complex-TaskAuctions) (Complex-TaskNegotiations) Table1.1: SystemFrameworksofOurTask-AllocationAlgorithms to robot r 1 , targets x 1 and x 4 are currently allocated to robot r 2 , and target x 3 is currently allocated to robot r 3 . Figure 1.12 illustrates one possible interac- tion sequence of robots if they follow the multi-party negotiation protocol to re- allocate targets among them. Robot r 1 first announces its allocated target x 2 to robotsr 2 andr 3 . Then, robotr 2 proposes the exchanges of targetsx 2 andx 4 and targetsx 3 andx 1 withitsevaluationtorobotr 3 . Afterthat,robotr 3 evaluatesthis proposalandsendstheupdatedproposaltorobotr 1 . Finally,robotr 1 acceptsthis proposalandnotifiesrobotsr 2 andr 3 toupdatetheirallocations. As tabulated in Table 1.1, we summarize the system frameworks of our task- allocationalgorithmsdevelopedinthisdissertationasfollows: • AuctionAlgorithmsforAllocatingTasks: Inthefirststageofthetask-allocation phase,wedevelopthetask-allocationalgorithmsthatimplementtheiterativeauc- tionprotocolforallocatingunallocatedtaskstorobots,calledauctionalgorithms. The auction algorithms for solving simple-task allocation problems are called simple-task auctions,whichallocateunallocatedsimpletaskstorobots. Theauc- tionalgorithmsforsolvingcomplex-taskallocationproblemsarecalledcomplex- taskauctions,whichallocateunallocatedsimpleandcomplextaskstorobots. • Negotiation Algorithms for Re-Allocating Tasks: In the second stage of the task-allocation phase, we develop the task-allocation algorithms that implement 23 Stage1: AllocatingTasks(AuctionAlgorithms) Stage2: Re-AllocatingTasks(NegotiationAlgorithms) Allocatingallunallocatedtaskstorobots Performingtaskexchangesamongrobots inoneroundsothattheresultingteam thatmayinvolveallallocatedtasks Optimality costissmallest,namely,thecurrentteam inoneroundsothattheresultingteam costincreasestheleast. costissmallest,namely,thecurrentteam costdecreasesthemost. Repeatedlyallocatingoneadditionalun- Repeatedlyperformingsingletaskexchangebetween MyopicOptimality allocatedtasktorobotsinoneroundso tworobotsinoneroundsothatthecurrentteam onSingleTasks thatthecurrentteamcostincreasesthe costdecreasesthemost,untiltheteamcostcan least,untilalltaskshavebeenallocated. nolongerbereducedbysingletaskexchanges. Repeatedlyallocatingk additionalun- Repeatedlyperformingk taskexchangesamong MyopicOptimality allocatedtaskstorobotsinoneroundso multiplerobotsinoneroundsothatthecurrent onMultipleTasks thatthecurrentteamcostincreasesthe teamcostdecreasesthemost,untiltheteamcost least,untilalltaskshavebeenallocated. cannolongerbereducedbyk taskexchanges. Table1.2: PerformancePrinciplesofOurTask-AllocationAlgorithms Stage1: AllocatingTasks(AuctionAlgorithms) Stage2: Re-AllocatingTasks(NegotiationAlgorithms) Optimality Optimality (NP-hard[CSS06]) (NP-hard[Dia04]) Simple-TaskAllocation MyopicOptimalityonSingleTasks MyopicOptimalityonSingleTasks (ExistingApproaches[TLJK05,LMK + 05]) (ExistingApproaches[San98,GMR97a]) MyopicOptimalityonMultipleTasks MyopicOptimalityonMultipleTasks (Chapter3) (Chapter4) Optimality Optimality (NP-hard[Ger03]) (NP-hard[Ger03]) Complex-TaskAllocation MyopicOptimalityonSingleTasks MyopicOptimalityonSingleTasks (Chapter5) (Chapter6) MyopicOptimalityonMultipleTasks MyopicOptimalityonMultipleTasks (FutureWork) (FutureWork) Table1.3: CategoriesofOurTask-AllocationAlgorithms eithertheone-to-oneorthemulti-partynegotiationprotocolforre-allocatingtasks amongrobots,callednegotiationalgorithms. Thenegotiationalgorithmsforsolv- ing simple-task allocation problems are called simple-task negotiations, which re-allocate simple tasks among robots. The negotiation algorithms for solving complex-taskallocationproblemsarecalledcomplex-tasknegotiations,whichre- allocatesimpleandcomplextasksamongrobots. 1.3.2 PerformancePrinciples In the previous subsection, we described how robots interact with each other in our task-allocation algorithms. However, we intentionally ignored how robots make their 24 allocation decisions. For example, we did not explain how robots bid on unallocated targets, how the auctioneer selects the winning bid, and why the auctioneer allocates only one target to some robot per round in Figure 1.7. In this subsection, we discuss the principles of our task-allocation algorithms for making allocation decisions, called performanceprinciples. Sincewewanttheresultingteamcostofourtask-allocationalgorithmstobeassmall as possible, the most desirable performance principle isoptimality: tasks are allocated torobotssothattheresultingteamcostissmallestinthefirststageofthetask-allocation phase, or tasks are re-allocated among robots so that the resulting team cost is smallest inthesecondstageofthetask-allocationphase. In the literature, there exist task-allocation algorithms that achieve the optimality performanceprinciple: • CombinatorialAuctions: Acombinatorialauction[CSS06]allocatesalltasksto robots in one round so that the resulting team cost is smallest. In a combinatorial auction, each robot evaluates and submits bids on all possible bundles (sets) of unallocated tasks, and the auctioneer evaluates all possible allocations of tasks to robots in order to determine the optimal allocations. (See Section 3.2 for a more detaileddescriptionofcombinatorialauctions.) • Combinatorial Exchanges: A combinatorial exchange [PKE01] re-allocates all allocatedtasksamongrobotsinoneroundsothattheresultingteamcostissmall- est. In a combinatorial exchange, each robot proposes and evaluates all possible taskexchangesthatinvolveitscurrentlyallocatedtasks,andtheleaderoftherobot teamevaluatesall possiblere-allocations oftasks amongrobots inorder todeter- mine the optimal allocations. (See Section 4.2 for a more detailed description of combinatorialexchanges.) 25 Although the optimal allocations with the smallest team costs are the most desir- able outcomes of any task-allocation algorithm, one cannot hope to always achieve the optimality performance principle in any realistic robot system since it is known to be NP-hard to solve task-allocation problems optimally. For example, combinatorial auc- tions are not efficient: 1) the number of bids submitted by each robot is exponential in the number of tasks; and 2) the auctioneer has to solve an NP-hard winner determina- tion problem [HB00] to determine the optimal allocations. Thus, it is expected that all task-allocation algorithms that achieve the optimality performance principle can solve onlysmalltask-allocationproblems. Given the difficulty of solving task-allocation problems optimally, it is natural to study heuristic task-allocation algorithms that produce sub-optimal allocations with small response times. In the literature, many researchers apply the idea of single-item auctions[TLJK05,LMK + 05]orsingle-itemexchanges[San98,GMR97a]totheirtask- allocation algorithms. In a single-item auction, only one task is auctioned off in each round,andthetaskisawardedtosomebiddersothatthecurrentteamcostincreasesthe least. In a single-item exchange, only one task exchange is performed in each round, and the task exchange is selected so that the current team cost decrease the most. 6 We formalize the performance principle of single-item auctions and single-item exchanges to be the myopic optimality (performance principle) on single tasks: one additional unallocated task is allocated to some robot in each round so that the current team cost increases the least, until all tasks have been allocated in the first stage of the task- allocation phase, or one task exchange is performed between two robots in each round 6 Actually, single-item auctions are the decentralized mechanism equivalent to a centralized greedy method which allocates a single task to the best robot at a time, and single-item exchanges are the decentralized mechanism equivalent to a centralized greedy method which performs local search [Lin65,LK73,WSB73]toswaptaskstoimprovethecurrentallocations. 26 so that the current team cost decreases the most, until the team cost can no longer be reducedbyanysingletaskexchangesinthesecondstageofthetask-allocationphase. Thetask-allocationalgorithmsthatachievethemyopicoptimalityperformanceprin- ciple on single tasks are promising for allocating tasks to robots quickly since robots needtoevaluateonlysingletaskallocationsorsingletaskexchanges. However,single- item auctions or single-item exchanges are too short-sighted and easy to get stuck in local optima and thus usually result in highly sub-optimal allocations [Dia04, ZKT06]. Forexample,ifthecurrentallocationscanonlybeoptimizedbytaskexchangesinvolv- ing more than two robots at a time, then single-item exchanges cannot reduce the team cost any more. In this dissertation, we propose the following performance principle of myopic optimality (performance principle) on multiple tasks to make our task- allocationalgorithmslessgreedy: k additionalunallocatedtasksareallocatedtooneor morerobotsineachroundsothatthecurrentteamcostincreasestheleast,untilalltasks havebeenallocatedinthefirststageofthetask-allocationphase,orktaskexchangesare performed among multiple robots in each round so that the current team cost decreases the most, until the team cost can no longer be reduced by up-tok task exchangesin the second stage of the task-allocation phase. Table 1.2 summarizes all three performance principles. Inthisdissertation,westudytask-allocationalgorithmsthatachievethemyopicopti- mality performance principle for solving both simple-task allocation and complex-task allocation problems. Table 1.3 summarizes the categories of our task-allocation prob- lems that are the combinations of the system frameworks shown in Table 1.1 and the performanceprinciplesshowninTable1.2. 27 NaiveApproaches OurApproaches Eachrobotbidsoneverynon-emptybundles Thenumberofbidsthateachrobotsubmitstothe ofatmostk tasksperround. auctioneerperroundisaconstant. Simple-TaskAuctions Theauctioneerevaluatesallpossible Theallocationsthattheauctioneerneedstoevaluate allocationsofk taskstorobots. islinearinthenumberofrobotsandindependentof thenumberoftargets. Simple-TaskNegotiations Eachrobotevaluatesandproposeseverypartial Eachrobotevaluatesandproposeseveryprofitable k-swapsthatinvolvesitself. partialk-swapsthatinvolvesitself. Eachrobotbidsoneverypossibleexecutiontimes Thereactionfunctionthateachrobotbidstothe ofitsallocatedcomplextasks(aninfinite auctioneercontainsafinitenumberofsegments. Complex-TaskAuctions numberofbids). Theauctioneerevaluatesallpossibleexecution Theexecutiontimesofcomplextaskstheauctioneer timesofcomplextasks. needstoevaluatearefinite. Complex-TaskNegotiations Eachrobotevaluatesandproposeseverypossible Thereactionfunctionthateachrobotproposescontains executiontimesofitsallocatedcomplextasks. afinitenumberofsegments. Table1.4: TheoreticalContributionsofOurTask-AllocationAlgorithms 1.4 OurContributions The main work of this dissertation is to develop auction and negotiation algorithms that achieve the myopic optimality performance principle (on either single or multiple tasks). This work is non-trivial since it is unknown how to design the local compu- tation and communication behaviors of robots in the system framework of the auction and negotiation protocols in order to achieve the given myopic optimality performance principle. Thisdissertationmakesthefollowingcontributions: Contributions to Simple-Task Auctions: In Chapter 3, we develop a new auction algorithm that implements the iterative auction protocol for allocating simple tasks to robotsinthefirststageofthetask-allocationphase. Thisalgorithmachievesthemyopic optimality performance principle on multiple tasks, namely, to allocate k unallocated taskstorobotsinoneroundsothatthecurrentteamcostincreasestheleast, where 1≤ k≤∞isauser-definedparameter. Ourgoalistospecifywhichlocalinformationrobots should send to the auctioneer and how the auctioneer should process this information efficientlyinordertodeterminethektaskallocationsthatincreasethecurrentteamcost theleastineachround. 28 We develop a novel data structure, called bid trees, from which each robot can con- structandsubmitbidstotheauctioneerindependently. Ourtheoreticalresultsshowthat the number of bids each robot submits is only a constant number for a given value of k. We prove that these bids succinctly characterize all necessary local information of robots needed by the auctioneer in order to determine the allocations that achieve the myopicoptimality performance principle onmultiple tasks. Wethen developawinner- determinationalgorithmfortheauctioneertodeterminesuchallocationswitharuntime thatisonlylinearinthenumberofrobotsandindependentofthetotalnumberoftasks. Ourexperimentalresultsshowthatournewauctionalgorithmcanallocatesimpletasks to robots efficiently and reduce the team costs of several existing auction algorithms significantly. ContributionstoSimple-TaskNegotiations: InChapter4,wedevelopanewnego- tiation algorithm that implements the multi-party negotiation protocol for re-allocating simple tasks among robots in the second stage of the task-allocation phase. This algo- rithm achieves the myopic optimality performance principle on multiple tasks, namely, to perform k task exchanges among robots in one round so that the current team cost decreases the most, where 1≤ k ≤∞ is a user-defined parameter. Our goal is to specify which local information robots should send to each other and how they should process this information efficiently in order to perform k task exchanges that decrease thecurrentteamcostthemostineachround. We develop a novel data structure, called partialk-swaps, that each robot can con- struct and propose to the other robots independently. Our theoretical results show that robots need to propose only profitable partialk-swaps to each other, which usually are only a small part of their local computation results. We prove that profitable partial k- swaps succinctly characterize all necessary local information of robots needed in order to perform k task exchanges that achieve the myopic optimality performance principle 29 on multiple tasks. We then develop a distributed algorithm for robots to construct and perform multiple task exchanges in each round so that the current team cost decreases the most. Our experimental results show that our new negotiation algorithm can re- allocate tasks among robots efficiently and reduce the team costs of the given initial allocationssignificantly. Contributions to Complex-Task Auctions: In Chapter 5, we develop a new auc- tion algorithm that implements the iterative auction protocol for allocating both simple and complex tasks to robots in the first stage of the task-allocation phase. This algo- rithm achieves the myopic optimality performance principle on single tasks, namely, to allocate one additional unallocated (either simple or complex) task to some robot(s) in oneroundsothatthecurrentteamcostincreasestheleast. Ourgoalistospecifywhich local information robots should send to the auctioneer and how the auctioneer should process this information efficiently in order to determine the single task allocation that increasesthecurrentteamcosttheleastineachround. We develop a novel data structure, called reaction functions, that each robot can construct and submit to the auctioneer independently. Our theoretical results show that robots need to submit only a finite number of execution times of complex tasks to the auctioneereventhoughthereareaninfinitenumberofpotentialexecutiontimesofcom- plex tasks. We prove that reaction functions succinctly characterize all necessary local informationofrobotsneededbytheauctioneertodeterminetheallocationdecisionsthat achieve the myopic optimality performance principle on single tasks. We then develop an approximation technique for robots to construct their reaction functions so that they need to submit only a constant number of execution times of complex tasks to the auc- tioneer. Our experimental results show that our auction algorithm can allocate both complex and simple tasks to robots efficiently and reduce the team costs of an existing auctionalgorithmsignificantly. 30 Contributions to Complex-Task Negotiations: In Chapter 6, we develop a new negotiation algorithm that implements the one-to-one negotiation protocol for re- allocatingbothsimpleandcomplextasksamongrobotsinthesecondstageofthetask- allocation phase. This algorithm achieves the myopic optimality performance princi- ple on single tasks, namely, to perform either a simple-task or complex-task exchange between two robots in one round so that the current team cost decreases the most. Our goalistospecifywhichlocalinformationrobotsshouldsendtoeachotherandhowthey should process this information efficiently in order to perform the single task exchange thatdecreasesthecurrentteamcostthemostineachround. Our theoretical results show that reaction functions also succinctly characterize all necessary local information of robots needed in order to perform the single task exchange that achieves myopic optimality performance principle on single tasks. We then develop a distributed algorithm based on reaction functions to construct and per- formeitherasimple-taskorcomplex-taskexchangethatdecreasesthecurrentteamcost the most in each round. Our experimental results show that our new negotiation algo- rithm can re-allocate simple or complex tasks among robots efficiently and reduce the teamcostsofthegiveninitialallocationssignificantly. Table1.4summarizesourtheoreticalcontributions. 1.5 DissertationStructure Thisdissertationisstructuredasfollows: Inthenextchapter,wediscussthebackground ofourapproachindevelopingthetask-allocationalgorithms. Wethendeveloptheauc- tion and negotiation algorithms for solving simple-task allocation problems in Chapter 3 and Chapter 4, respectively. Next, we develop the auction and negotiation algorithms 31 for solving complex-task allocation problems in Chapter 5 and Chapter 6, respectively. WefinallyconcludeinChapter7. 32 Chapter2 Background Traditional artificial intelligence techniques focus on the abilities of individual entities, suchassearch,planningandlearning[RN03]. However,therecentfastdevelopmentof multi-robotsystems[FIN04]shiftsresearchers’attentiontodistributedproblemsolving [DR94, Dur99, Les02], namely, how to make multiple robots coordinate and cooperate with each other in order to accomplish team missions in a distributed fashion. As dis- cussedinSection1.1.2,solvingtask-allocationproblemsisoneofthekeyissuesindis- tributedproblemsolving. Inthischapter,wediscussthebackgroundofourauctionand negotiation algorithms developed in this dissertation for solving task-allocation prob- lems. We begin this chapter with an overview of different categories of task-allocation problems in the literature. After that, we summarize several existing approaches of developingtask-allocationalgorithms. 2.1 Problems It is widely believed that a team of robots has several advantages over single robots: i) multiple robots can perform tasks in parallel and thus improve the effectiveness of a robotic system; ii) multiple robots can perform complex tasks that single robots cannot do and thus improve the capabilities of a robotic system; and iii) multiple robots are moreresilienttofailureandthuscanimprovetherobustnessandreliabilityofarobotic system. In the past decade, robot teams have been proposed and utilized in a variety of 33 settings,accomplishingdifferentgoals,andappliedinmanyapplicationdomains. How- ever,itisnon-trivialtocoordinatemultiplerobotswellinadistributedfashioneventhat robots are all cooperative [BMBB09] since they still have to determine which robots shouldexecutewhichtasksandwhenshouldtheyexecutethetaskssothattheteamper- formance is as high as possible. It is critical that multi-robot systems are able to solve task-allocation problems efficiently and effectively. In this section, we overview differ- ent categories of task-allocation problems studied in the literature in terms of different application domains, different robot models, different task models, and different team objectives. 2.1.1 ApplicationDomains A wide range of real-world applications of multi-robot systems have to solve task- allocation problems for coordinating multiple robots. In the following, we give some examplesofsuchapplicationdomains: • Multi-Robot Routing: In a multi-robot routing problem, a team of robots is coordinated to visit multiple locations in a terrain [LMK + 05]. Examples includesearchandrescue[KTN + 99,RFMJ10,NSK + 05], Mar-roverrockprobes [Ker90,TONG + 02],anddisasterresponse[JDS07,RHL10]. • Multi-RobotExploration: Inamulti-robotexplorationproblem,ateamofrobots iscoordinatedtoexploreanunknownenvironment[BMF + 00]. Examplesinclude planetary exploration [Moo01, MS01], reconnaissance [WA04, HBB + 00], and map-building[LML09,SAB + 00,LLW05]. 34 • Multi-Robot Observation: In a multi-robot observation problem, a team of robots is coordinated to detect and track a set of moving targets [Par99]. Exam- ples include civil security [EGHL93, LTC10, BM03a], military surveillance [EKK12,EAK09,DC11],andtargettracking[JS06,WG12,ZR11]. • Multi-Robot Coverage: In a multi-robot coverage problem, a team of robots is coordinatedtocoverallfreespacesintheenvironment[Cho01,ZKKJ10]. Exam- ples include de-mining [LGMV05, EK12, Cas00], lawn mowing [HH00, CD11], andcar-bodypainting[SWN91,AGC + 05]. • Multi-RobotScheduling: In a multi-robot scheduling problem, a team of robots is coordinated to accomplish jobs or share resources in order to minimize inter- ference and/or maximize throughput [Bru01]. Examples include robot auto- maticrecruitment[Gag04,MK00,POCD09],loadbalancing[MSR + 03,GQC05, NK11],andmulti-purposeprocessorscheduling[Cyb89,SBGC07,ZP13b]. • Multi-RobotControl: Inamulti-robotcontrolproblem,ateamofrobotsiscoor- dinated to interact with each other to perform tasks with constraints [Mat92]. Examples include box pushing [RDJ95, MNS95, WS06, GTP08], object trans- portation[SSH + 01,BBD + 97],andinfrastructureconstruction[BML + 00,LGJ01, WSM02]. In this dissertation, we use multi-robot routing as examples to explain our task- allocation algorithms and also as a standard testbed for evaluating them empirically. Many task-allocation algorithms [PR95, KS08, LMK + 05, VKP10, MML08, MML09, EKK09, Hea13] use multi-robot routing as example problems since they are easy to illustrate and simple to understand. Multi-robot routing problems with simple targets are also similar to multiple travelling salesman problems [Bek06] that have been well studiedintheoreticalcomputerscienceandoperationresearchexceptthatrobotsdonot 35 necessarily start at the same location and not required to return to their start locations. As a result, we are able to formulate multi-robot routing problems as a mixed integer programandsolvethemoptimally(forthosewithsmallproblemsizes),whichallowsus tobettercomparetheperformanceofourauctionandnegotiationalgorithmsdeveloped inthisdissertationempirically. However,itisimportanttokeepinmindthatourauction andnegotiationalgorithmscanalsobeappliedtootherdomainsaswell. 2.1.2 RobotModels A team of robots is provided to execute tasks in a task-allocation problem. There are different types of robots in multi-robot systems. In this subsection, we characterize the mostcommoncategoriesofrobotmodelsintheliteratureasfollows: • Robot Capability: If all robots in the team have the same capability, then we call them homogeneous robots; Otherwise, we call them heterogeneous ones. Robot systems with homogeneous robots have the advantage that they are easy todesignandreplace,andthustheyarecheaperandmorescalable[LK10]. Many swarm-based robot systems [Yam97, FM01, SMS06, YSS + 07, KK11] consist of homogeneous robots and many task-allocation algorithms [DMS03, TLJK05, LMK + 05,SBE09]alsosimplifytheirrobotstobehomogeneousintheearlystage of development. However, the increasing complexity of tasks in robot systems requires for more specialized capabilities of robots [Par94, INPS03]. Thomas and Williams [TW09] extend sequential single-item auctions [TLJK05] to het- erogeneous robots. Michael et al. [MZKP08] first study coalition formation for homogenous robots, and then Fiaschi and Pacini [FP11] generalize their work to heterogenous robots. Kernbach et al. [KSH + 11] discuss how to balance the degree of heterogeneity in a multi-robot system. In this dissertation, we assume thatrobotshavedifferentcapabilitiesinourtask-allocationalgorithms. 36 • Robot Capacity: If each robot in a task-allocation problem is capable of exe- cuting at most one task, then the task-allocation problem is an optimal assign- ment problem [Gal89]. The optimal assignment problem can be solved by the Hungarian method [Kuh55] optimally in O(n 3 ) time, where n is the number of robots. Researchers [GM02, VMO07, VH07, SB06] have also developed auc- tion algorithms for solving optimal assignment problems greedily. Sariel et al. [SB06,SBE06]applytheoptimalassignmentproblemtomulti-robotexploration. Viguriaetal. [VH07]provideanupper-boundanalysisfortheresultingteamcosts of these auction algorithms. However, it is more general and practical in a robot system that robots are capable of executing multiple tasks. For example, a robot can be allocated and then locally schedule its route to visit multiple locations in multi-robot routing. The majority of existing task-allocation problems studied in the literature belong to this category. However, the computational complexity of a task-allocation problem increases significantly when robots can execute multi- pletasks. Forexample, solving multi-robot routing problems is knownto be NP- hard[LMK + 05,Sur10,Sit02]. Thus,researchersdevelopheuristictask-allocation algorithms in this category in order to allocate tasks efficiently. In this disser- tation, we assume that robots can execute multiple tasks in our task-allocation algorithms. • Robot Communication: Multi-robot systems can be classified into the follow- ing three categories based on the communication among robots: i) full commu- nication among robots; ii) limited communication among robots with range and topology constraints; and iii) no explicit communication among robots. Firstly, many existing task-allocation algorithms assume that robots can explicitly com- municate with each other in order to exchange local information and make con- sistent decisions. Secondly, several researchers study task-allocation algorithms 37 for robots with communication range constraints [CBH09, BM03b, MML08] or communication topology constraints [RB04, ZSJP08, WS13]. Since a robot can communicate only with a sub-group of the team given these constraints, robots may make conflicting allocations caused by the inconsistent informa- tion among them [FM04]. As a result, the task-allocation algorithms usu- ally make the robots use some consensus algorithms [SM04, MR06, OT09] to converge on consistent allocations. Finally, many swarm-based robot systems [Iba96, Mac91, FM01, Yam97, Par93, BH00] do not require robots to explicitly communicatewitheachother. Robotsinthesesystemsusuallyobeypre-designed operationalrules,andthentheteamlevelcooperativebehaviorsemergefromtheir rule-based interactions [Ark98]. In this dissertation, we assume that robots can fullycommunicatewitheachother. • Robot Failure: One of the main advantages of using multi-robot systems is that the redundancy of robots increases the robustness and fault tolerance of the system [Bro86]. Parker [Par98] propose to use the impatience motiva- tion for robots to monitor and take over tasks that the other robots fail to execute. Stancliff and Dolan [SD10] demonstrate that the robot team perfor- mance can be greatly improved by using the knowledge of expected robot fail- ure rates when making initial task allocations. Kaminka et al. [KT97] utilize social comparison theory [New94] to detect unpredictable robot failures in a multi-robot system. However, many existing task-allocation algorithms do not explicitly consider robot failures. They either assume that robots do not fail [TLJK05, GC06, CGG13, SA10, Hea13] or repeatedly run the task-allocation algorithms to allocate the tasks that are not yet executed [NG10] or previously allocated to the failed robot [HK05] to the remaining robots. In this dissertation, weassumethatrobotsdonotfailinourtask-allocationalgorithms. 38 2.1.3 TaskModels In this subsection, we discuss different assumptions and constraints on tasks studied in theliteraturefortaskallocation. • CostSynergy: Whenarobotcanexecutemultipletasksinatask-allocationprob- lem, these tasks are said to have synergy for the robot if the cost of the robot to execute them together is different from the sum of its costs of executing them individually. Many multi-robot scheduling problems [HLP93, OHB + 98, ZP13b] andcoalitionformationproblems[SK98,VA06,Su07]studytaskswithoutsyner- giesamongthem. Forexample,theprocesstimesofrobotsforexecutingtasksin job-shop scheduling [GJS76] and the utilities of coalitions for executing tasks in [SK98, VA06, Su07] are fixed and independent of the tasks that are already allo- cated to robots. However, it is more common that tasks have synergies for robots in a task-allocation problem. For example, in multi-robot routing [LMK + 05] the route distance of a robot to visit multiple locations is usually different from the sum of the route distances of the robot to visit these locations individually. Simi- larly,inresourceallocations[BGS99]theutilityofarobottoobtainoneadditional resourceoftendependsontheircurrentlyallocatedsubstitutableandcomplemen- taryresources. Inthisdissertation,weassumethattaskshavesynergiesforrobots inourtask-allocationalgorithms. • Precedence/Temporal Constraints: Tasks may have precedence constraints among them, namely, a task must be executed before or after some other tasks [CL87, PS89, JDS08]. Vidyarthi and Tripathi [VT96] propose an optimal policy forallocatingtaskgraphstorobotsthatarecharacterizetheprecedenceconstraints oftasks. Ahmad and Kwong[AK98] developseveralsearch algorithms based on A ∗ [Nil82] to allocate tasks with a guaranteed bound from the optimal solution. 39 Luo et al. [LCS11] study task-allocation problems where tasks are divided into groupsandthereareprecedenceconstraintsbetweengroups. Tasksmayalsohave temporal constraints on them, namely, a task must be executed in a given time window [Bra02, NNG12] or before a certain deadline [GS09, ZCL12, LCS13]. Jones et al. [JDS07] develop a learning-enhanced task-allocation algorithm for oversubscribed domains where robots cannot execute all tasks before deadlines. Kong et al. [KZYL13] propose a negotiation algorithm for robots to re-allocate tasks with different time constraints in grid environments. Zhao et al. [ZCL12] use a variable neighbourhood search approach to solve multiple traveling sales- manproblemwithdeadlines[Bek06]. Inthisdissertation,weassumethattasksdo nothaveanyprecedenceortemporalconstraintsinourtask-allocationalgorithms. • Complex Task: A task is called complex in a task-allocation problem if it has to beexecutedbymorethanonerobot[ZS05,Zlo06,MDKJ06,ZK08,ZK11]. There exist several different categories of complex tasks in the literature. One example is that a complex task can be represented in a tree structure with AND/OR logic operatorsconnectingwithsubtasks[SA98]. Zlotetal. [ZS05,Zlo06]developtask treeauctionsthatmakerobotsbidtheirevaluationsondifferentdecompositionsof atree-likecomplextask. ZhangandParker[ZP13a]studycomplex-taskallocation problemswheretaskshaveinter-taskresourceconstraints. Khamisetal. [KEK11] explore both centralized and distributed algorithms of allocating decomposable surveillance tasks to mobile robots. In this dissertation, we study a different type ofcomplextasks,namely,acomplextaskneedstobeexecutedbymultiplerobots simultaneously. Suchcomplextasksarecommoninreal-worldapplications. For example,largefirescanonlybeextinguishedwithseveralfireengines,andheavy objectscanonlybemovedwithseveralrobots. 40 • Static/Dynamic Environments: If all tasks are already known before the task- allocation algorithm is executed, then the task-allocation problem is in a static environment. Otherwise, new tasks are introduced over time and the task- allocation algorithm has to continuously allocate newly coming tasks to robots, thenthetask-allocationproblemisinadynamicenvironment. Severalresearchers directly apply the task-allocation algorithms developed in static environments to dynamic environments. For example, Schoenig and Pagnucco [SP11] extend sequential single-item auctions to dynamic environments. Heap et al. [HP13] repeatedlyrunsequentialsingle-clusterauctions[HP11,Hea13]toallocateonline tasks with pickup and delivery requirements. Luo et al. [LCS12] analyze com- petitive ratios of greedy auction algorithms for solving task-allocation problems in dynamic environments where robot can execute at most one task at a time. However,themajorityofthetask-allocationalgorithmsfordynamicenvironments make use of ideas from queueing theory [BDP03] and control theory [Bro74] studied in operations research. For example, Melndez et al. [MDL12] propose a stochastic model for robots to detect randomly distributed landmines based on the estimation of a stochastic queue of pending demands. Smith [Smi09] devel- ops stochastic algorithms with worst-case performance guarantees for dynamic vehicleroutingbycombiningideasfromcombinatorialoptimizationandstochas- tic queueing theory. Tsalatsanis et al. [TYV09a] make use of limited lookahead policiesintotraditionalsupervisorycontroltheorytocoordinateateamofmobile robots to patrol a warehouse containing hazardous materials. In this dissertation, weassumethattasksarealwaysknowntorobotsbeforeourtask-allocationalgo- rithmsareexecuted. 41 2.1.4 TeamObjectives The robot team is coordinated to execute tasks to maximize the team performance for some team mission. The optimization depends on the team objective specified in the givenmission. Inthissubsection,wedescribethemostcommonteamobjectivesstudied intheliteratureforcooperativerobots. 1 • Minimize Total Robot Costs: This team objective is called the MiniSum team objective in this dissertation. In a task-allocation problem with this team objec- tive, tasks are allocated to robots so that the sum of robot costs of all robots is as small as possible. For example, a multi-robot routing problem with the MiniSum teamobjectiveisaspecialcaseofmultipletravellingsalesmanproblems[Bek06] (except that robots do not need to return to their start locations) that are trying to minimize the total travel distances of all robots. Existing task-allocation algo- rithms studied for this team objective include [TLJK05, ZKT06, WD13, LAL04, TYV09b,CNC12]. • MinimizeMaximalRobotCost: ThisteamobjectiveiscalledtheMiniMaxteam objectiveinthisdissertation. Inatask-allocationproblemwiththisteamobjective, tasksareallocatedtorobotssothatthemaximalrobotcostisassmallaspossible. For example, many job-shop problems and multi-robot scheduling problems are tryingtominimizethetask-completiontime(makespan). Existingtask-allocation algorithms studied for this team objective include [TLJK05, ZKT06, ZKKJ10, JPM04,Hag96]. 1 In this dissertation, robots are cooperative and always collaborate to maximize the team perfor- mance. Thus, we do not include optimization objectives for self-interested robots such as Pareto opti- mality[HR69]intodiscussionhere. 42 • MinimizeTotal TaskLatencies: This team objective is called the MiniLat team objective in this dissertation. In a task-allocation problem with this team objec- tive,tasksareallocatedtorobotssothatthesumofexecutiontimesofalltasksis assmallaspossible. Forexample,amulti-robotroutingproblemwiththeMiniLat teamobjectiveisaspecialcaseofmultiplerepairmanproblems[FHR03,FHR07] (except that robots do not need to return to their start locations) that are trying to minimizethetotallatenciesofalltasks. Existingtask-allocationalgorithmsstud- iedforthisteamobjectiveinclude[ILR04,TLJK05,ZK10,HKKK13,LQL14]. • MaximizeTotalThroughput: Thethroughputofarobotsystemisdefinedtothe number of tasks executed by robots [HP07]. In a task-allocation problem with thisteamobjective,tasksareallocatedtorobotssothatasmanytasksaspossible canbeexecutedbyagivennumberofrobotswithinatimeperiod,orequivalently, as few robots as possible are required to execute a given number of tasks within a time period. Existing task-allocation algorithms studied on this team objective include[BCF + 01,BBLM10,Wei00,CL07,VKP10,KHS12]. Inthisdissertation, wedonotstudythisteamobjectiveforourtask-allocationalgorithms. 2.2 Approaches Inthissection,weoverviewdifferentapproachestosolvingtask-allocationproblemsin theliterature. 2.2.1 CentralizedApproaches Inacentralizedrobotsystem,asinglerobot(orthecentralplanner)actsofthe”leader” oftheteamandisresponsibleforsolvingthetask-allocationproblemfortheentireteam 43 [Dia04]. All other robots in the team continuously convey their relevant local informa- tiontotheleader,andthenexecutethetasksallocatedbytheleader. Forexample,Sycara et al. [SDP + 96] develop a flexible architecture called RETSINA for supporting a col- lection of distributed robots to cooperate with each other asynchronously to perform a varietyoftasks. InRETSINA,atask-specificcomponentreceivestasksfromend-users, decomposesthemintosub-tasksbasedonitsdomainknowledge,andthenallocatesthe sub-tasks to robots. This task-specific component takes full control for collecting data, resolvingconflicts,allocatingtasksandcoordinatingtherelatedrobots. Anothercentral- izedapproachexampleistheworkdonebyBrumittandStentz[BS96]forcoordinating multiple robots to follow the optimal paths to visit several targets in initially unknown terrain. They develop a centralized component, called ”dynamic planner”, that uses the D ∗ algorithm [Ste94] to continuously update the paths for all robots, while each robot provides its current local terrain information to the dynamic planner and then follows thepathselectedbythedynamicplanner. Otherexistingcentralizedalgorithmsinclude [SAF + 00,RM99,MCM99,CSKC01,CSKC01,ZJKK05]. Many researchers use optimization techniques developed in operations research for the leaders of the centralized systems in order to solve task-allocation problems opti- mally or near-optimally. These algorithms can be classified based on the methods they useasfollows: • Branch-and-Cut: The task-allocation problem can be modeled as a mixed inte- ger program (MIP) [Sch98] and then be solved by a LP/MIP-solver, such as CPLEX [Cpl01] or SOPT [Sop07], to get the optimal solution. Branch-and- cut methods use the cutting-plane techniques to solve the MIP as follows: the LP/MIP solver first relaxes some constraints of the original MIP and then finds the optimal solution of the new MIP. If this solution satisfies all constraints of the original MIP, then it is also the optimal solution of the original MIP. 44 Otherwise, the LP/MIP solver adds the violated constraints to the current MIP and starts the process again until it finds the optimal solution of the origi- nal MIP. For example, Laporte et al. [LND85] develop an optimal algorithm for solving capacitated vehicle routing problems based on the relaxation of the capacity constraints, the distance constraints, and the sub-tour elimination con- straints. Baldacci et al. [BHM04] describe an optimal branch-and-cut algorithm based on a two-commodity network flow approach for solving the vehicle rout- ing problems where vehicles have the same capacity and depart from a central depot. Other existing centralized algorithms based on branch-and-cut include [CH93,PR91,RKPT03,Ral03,Fle88,ABB + 95]. • Branch-and-Bound: The task-allocation problem can be modeled as a search problem and then be solved by search techniques, such as depth-first search or A ∗ [Nil82], to get the optimal solution. Branch-and-bound methods calculate andupdatethelower/upperboundsoftheoptimalsolutionduringthesearchpro- cess and the optimal solution is found when the lower and upper bounds meet. For example, Kafil and Ahmad [KA98] reduce the search tree of A ∗ by prun- ing the nodes whose team costs are already larger than that of a randomly gen- erated solution. Since the effectiveness of branch-and-bound methods depends on the quality of the lower bounds they can provide, researchers study different approaches on computing the bounds. For example, Christofides et al. [CMT81] compute the lower bounds based on Lagrangian relaxation and Fisher [Fis94] computesthelowerboundsbasedonageneralizationofspanningtrees,calledM- trees. Other existing centralized algorithms based on branch-and-bound include [YFK10,MCC04,FTV94,PK99,Fuk09]. 45 • Set-Partitioning: The task-allocation problem can be modeled as a set- partitioning problem and then be solved by set-partitioning techniques, such as linearrelaxationandcolumngeneration,togettheoptimalsolution. Forexample, BalinskiandQuandt[BQ64]usetheset-partitioningformulationtosolvecapaci- tatedvehicleroutingproblems. Hadjiconstantinouetal. [HCM95]describealin- ear relaxation method for finding the optimal solution of the set-partitioning for- mulation based on the computation ofk-shortest paths. Baldacci et al. [BCM08] present an optimal algorithm for solving the capacitated vehicle routing prob- lembasedonset-partitioningwithadditionalcutsthatcorrespondtocapacityand cliqueinequalities. Otherexistingcentralizedalgorithmsbasedonset-partitioning include[SK95,AMS89,PRBL06,BSL97]. As discussed in [Dia04], the main advantage of centralized approaches is that their solutions are optimal or close to the optimal. However, these approaches are usually undertheassumptionthattheinformationconveyedfromrobotstotheleaderisaccurate and timely. Also, the leaders in these centralized systems are usually the computation bottleneckandthesinglepointoffailureofthesystem. 2.2.2 DistributedApproaches Inadistributedrobotsystem,robotsuselocalinformationtomakedecisionswithoutthe need of a central leader. Distributed task-allocation algorithms can be divided into two categoriesbasedonwhetherrobotsexplicitlycommunicatewitheachotherintherobot system. • Distributed Approaches without Communications: In the literature there are two basic approaches of coordinating distributed robots [FIN04], namely, emer- gentcooperationandintentionalcooperation. 46 – In an emergent cooperation system, robots may not know the existence of othermembersandtheyobeypre-designedoperationalrules. Theteamlevel cooperative behaviors emerge from the rule-based interactions of robots [Ark98]. Thus, emergent cooperation belongs to the category of the dis- tributed approaches without explicit communications among robots. For example, Chantemargue and Hirsbrunner [CH99] use stigmatic mechanism tocoordinateateamofrobotstocollectingobjects. Passino[Pas02]designs a non-gradient optimization algorithm based on social foraging behaviors [OBE90] for controlling autonomous vehicles. Other important emer- gent cooperation algorithms include [Iba96, Mac91, FM01, Yam97, Par93, BH00]. As noted in [FIN04], these emergent cooperation systems mainly focus on how to execute all tasks successfully instead of how to execute themefficientlyandeffectively. – Inanintentionalcooperationsystem,robotsareawarethepresenceofother team members and act together to accomplish the team goal. ALLIANCE [Par98] is an example of intentional cooperation in the category of the dis- tributed approaches without communications: robots make their own deci- sionsbasedontwomotivations,namely,impatienceandacquiescence 2 . The designofthemotivationalbehaviorsinALLIANCEallowsrobotstocooper- atewitheachothertoexecutealltaskswithouttheneedofexplicitcommuni- cationsamongthem. Similarly,AgassounonandMartinoli[AM02]develop threshold-based algorithms for allocating tasks to robots where each robot has an activation threshold for each task. Another example of distributed 2 When a new task is announced or some robot fails to execute a task in ALLIANCE, each robot increases its impatience at different speeds (that depends on its capability for the task). The robot whose impatience first exceeds a limit is allocated the task. After a robot is allocated a task, its acquiescence increases at certain speed until the task is executed. The robot may give up the task if its acquiescence exceedsalimit. 47 approaches without communications is developed by Dahl et al. [DMS03], where they make use of the concept of vacancy chains [CWD88] studied in social structure to allocate tasks. Robots in the vacancy chain algorithm does not need to communicate with each other since they calculate the task rewardsandmakedecisionscompletelybasedontheirlocalinformation. • DistributedApproacheswithCommunications: Themajorityofthedistributed taskallocationalgorithmsintheliteratureuseexplicitinter-robotcommunications to coordinate robots. These algorithms can be further categorized based on the contentsofinter-robotcommunications: – Tokens: In a token passing system [SFOT04], robots receive and pass a token to coordinate their actions and only the robot currently holding the tokencanmakedecisions. Forexample,Farinellietal. [FINZ06]usetokens torepresenttasks. Whenarobotreceivesatoken,itdecideswhethertoexe- cute the task in the token or pass the token to the next robot. This approach guarantees conflict-free allocations even when tasks are dynamically gener- ated. Xu et al. [XSY + 05] develop another token-based algorithm for allo- cating tasks to a large number of robots. They use token to encapsulate all information,calledcontrolinformation,thatneedstobesharedbytherobot team. RobotsthenlocallyapplyapartiallyobservableMarkovdecisionpro- cess(POMDP)[LCK95]todeterminewhenandwheretopassthetokens. – Situation Awareness: In a distributed system, robots may have inconsis- tentsituationawarenessontheirenvironment. Inordertomakeconflict-free allocations, many distributed algorithms require robots to share their situa- tion awareness within the team to resolve conflicts. Choi et al. [CBH09] 48 develop a consensus-based algorithm for robots to make consistent situa- tion awareness consistent before tasks are allocated to robots. They show thattheiralgorithmcanguaranteetheconvergenceofsituationawarenessfor many different network topologies. Alighanbari et al. [AH05] study how to efficiently synchronize different situation awareness among multiple aerial vehicles. Ifthevehicleswanttohavetheperfectglobalsituationawareness, theyhavetoexchangealotofinformationabouttheenvironment,theircur- rentstates,andtheirfutureintentions. Inordertoreducethecommunication efforts of robots, their algorithm proposes to use a flexible degree of syn- chronizationandalsointroducesasecondplanningsteptoaccommodatethe partiallyupdatedsituationawareness. – PartialSolutions: Insteadofexchangingtheirlocalinformation,robotscan update and prorogate partial solutions of the task-allocation problem with eachotheruntilthesolutionconvergestobestable,namely,thesolutiondoes not change any more over time. Bhattacharya et al. [BKL10] solve multi- robot path planning problems with pairwise constraints between robots in multiple cycles. In each cycle, each robot evaluates paths submitted from theotherrobotsandrevisesitsownpathtominimizetheresultingpathcost. The algorithm terminates when all paths become stable. Modi [Mod03] developsanasynchronousdistributedalgorithm,calledADOPT,forsolving distributed constrain optimization problems. Each robot in ADOPT locally decides the value of its own variable and propagates its decision with some cost estimations to the other robots. ADOPT allows each robot to execute asynchronouslyandisguaranteedtofindanoptimalsolution. – Bids/Proposals: In a market-based system, robots submit bids on tasks to theauctioneerinauctionalgorithms,ormakeproposalsofexchangingtasks 49 to the other robots in negotiation algorithms. Since we develop auction and negotiation algorithms in this dissertation, we give a detailed summary of existingmarked-basedapproachesinthenextsubsection. 2.2.3 Market-BasedApproaches Economics research has explored auctions and negotiations for many years on how to efficiently distribute resources to self-interested individuals [MM87]. Market- based approaches make use of the interaction framework of auctions and negoti- ations to allocate tasks to robots. The idea starts from the contract net protocol [Smi79, DS83, SD81], as discussed in Subsection 1.3.1. The contract net proto- col introduces a publish/subscribe communication model [Ger03] for robots to allo- cate and re-allocate single tasks through negotiations. AI research has applied the contract net protocol to many domains, such as sensor networks [SD78, SD80], industrial manufacturing systems [Dyk87, Bak88, SKB97, OHB + 98, SS04], trans- portation [CBER00, CD05, CDGP05, FKMP95], and distributed knowledge networks [YHH + 98, HPX07]. Most of these pioneer works are from an experimental perspec- tive [Dia04]. Recently, researchers have extended the original contract net protocol to market-inspired protocols, namely, auction and negotiation mechanisms, to allocate tasks and there has been progress on the formal analysis of market-based approaches formulti-robotcoordinationinthelastdecade. Forexample,Diasetal. [DZKS06]dis- cuss when it is appropriate to apply market-based approaches in a multi-robot system, and Lagoudakis et al. [LBK + 04] provide the first analytical performance guarantee of auction-based algorithms 3 . In this section, we give an overview of the existing auction andnegotiationalgorithmsintheliteraturefortaskallocation. 3 Theyshowthatsequentialsingle-itemauctions[TLJK05]inoneparticulardomain(multi-robotrout- ing)canresultinteamcoststhatareatmosttwotimeslargerthantheminimalonefortheMiniSumteam objective. 50 AuctionAlgorithms One pioneering work of extending the original contract net protocol to auction algo- rithms is TRACONET developed by Sandholm [San93]. The original contract net pro- tocoldoesnotspecifythedetailsofhowrobotsshouldbidontasksandhowtheauction- eershoulddetermine thewinningbids. Sandholmformalizesthe biddingandawarding decision processes for robots based on marginal cost calculations, namely, a robot bids on a task the increase in its robot cost for executing all tasks allocated to it if it is addi- tionally allocated the task, and the auctioneer allocates the task to the robot with the smallest bid. This formalization actually achieves the myopic optimality performance principleonsingletasksthatweintroducedonpage27ofChapter1. Later,Gerkeyand Mataric [GM02, GM04] propose an auction protocol based on the original contract net protocol in MURDOCH where each auction consists of five steps: task announcement, metric evaluation, bid submission, close of auction, and finally progress monitoring. This auction protocol is very similar to the iterative auction protocol shown in Fig- ure 1.5 except that we ignore the progress-monitoring step in this dissertation. Similar to TRACONET, a task in MURDOCH is also greedily allocated to the robot with the lowestbidcost. However,onlyidlerobotsinMURDOCHareeligibletoparticipantany auctionandthuseachrobotcanonlybeallocatedatmostonetaskatanytime. Sarieland Balch[SB05,SB06]developseveralsimilarauctionalgorithmsinthedomainofmulti- robot routing. In these algorithms, tasks are also allocated to robots in multiple rounds and thus each robot can be allocated at most one task at any time. They propose and evaluate different evaluation functions for robots to bid on tasks. One of them is called closest-cost where robots bid their distances from targets and the auctioneer chooses the smallest bid. Thus, their algorithms with the closest-cost evaluation functions also achieve the myopic optimality performance principle on single tasks under the premise thateachrobotcanbeallocatedatmostonetaskatanytime. 51 Dias and Stentz [DS00, Dia04] develop a distributed market-based system, called TraderBots, where tasks are allocated to robots in a dynamic environment. Unlike pre- viousalgorithms,eachrobotinTraderBotscanbeallocatedmultipletaskssothatrobots needtoschedulealocalplantoexecutetheirallocatedtaskstoimproveefficiency. Tovey et al. [TLJK05] develop sequential single-item auctions that also allocate tasks in mul- tiple rounds and one task is allocated to one robot per round. Robots in sequential single-item auctions can also be allocated multiple tasks. Sequential single-item auc- tions generalize the bidding rules of robots and the winner determination procedure of the auctioneer so that the allocation made by the auctioneer in each round always increases the team cost the least for different team objectives. Mosteo and Montano [MM07] perform comparative experiments of sequential single-item auctions for the MiniMaxandMiniSumteamobjectives. TheyindicatethatpursuingtheMiniMaxteam objective (i.e. minimizing the task-completion time) can also result in small team costs for the MiniSum team objective (i.e. minimizing energy consumptions), however, pur- suing the MiniSum team objective may produce large team-completion times for the Minimax team objective. Later, Sujit and Beard [SB07] apply sequential single-item auctionstocoordinatemultipleunmannedaerialvehiclestoperformsearchandsurveil- lance tasks. Schoenig and Pagnucco [SP11] apply sequential single-item auctions to dynamic environments where tasks are generated over time. Mosteo et al. [MML08] applysequentialsingle-itemauctionstorobotsthathavelimitedcommunicationranges. Kishimoto and Sturtevant [KS08] study and compare several different path planning techniqueswhenapplyingsequentialsingle-itemauctionstomulti-robotrouting. All previouslydiscussed algorithms are based on single-item auctions where robots bidonsingletasksandtheauctioneeronlyallocatesonetaskineachround. Theadvan- tageofsingle-itemauctionsisthatitisverysimpleandfasttoallocatetasks. However, the resulting team costs of single-item auctions can be highly sub-optimal, especially 52 when there are synergies among tasks [DS03, ZKT06]. Therefore, several researchers develop auction algorithms based on combinatorial auctions [CSS06]. Das and Grosu [DG05]makeeachrobotbidoneverypossiblecombinationofresourcesinagridcom- puting environment and use an approximate algorithm to solve the NP-hard winner determination problem. M. Berhault et al. [BHK + 03] explore several bidding strate- gies of robots in combinatorial auctions and compare their performance empirically in the domain of multi-robot routing. Rassenti et al. [RSB82] apply combinatorial auc- tions to allocate airport time slots to airlines in an airport management system. Edalat et al. [EXR + 11] apply combinatorial auctions to allocate multiple concurrent target tracking tasks to sensor nodes in distributed sensor networks. Walsh et al. [WWY00] applycombinatorialauctionstoallocatehierarchicaltaskstoproducersforsupplychain formation. Akbarimajd et al. [AAMH14] apply combinatorial auctions to coordinate multiplerobotstotransportgoodstodifferentlocations. Combinatorialauctionsresultinsmallerteamcoststhansequentialsingle-itemauc- tions [DS03], however, they are not efficient since robots have to evaluate and submit an exponential number of bids to the auctioneer in combinatorial auctions. In order to reduce the number of bids in combinatorial auctions, many researchers use clustering algorithmstogrouptasksintoclustersandthenmakerobotsbidonthesepre-determined clusters instead of all possible bundles of tasks. Elango et al. [ENT11] use K-means clustering techniques [HW79] to group tasks with the objective to minimize the total distances between tasks in each cluster. Olivares et al. [OWEY12] group school loca- tionsintoclustersiftheyarewithinacircularperimeterofauser-definedradius. Greene and Hofmann [GH06] use constrained clustering techniques [THLN01] to group new tasks with the existing tasks so that robots can execute the new tasks without dropping theirallocatedtasks. Berhaultetal. [BHK + 03]useanapproximatedmaximalgraph-cut algorithm [BYZ98] to group tasks. Heap and Pagnucco [HP12, Hea13] introduce the 53 clusterbiddingstrategyintosequentialsingle-itemauctionsandextendthemtosequen- tialsingle-clusterauctions. In the literature there exist several definitions of complex tasks, as we discussed in Subsection 2.1.3. Zlot et al. [ZS05, Zlo06] develop task tree auctions for allocat- ing complex tasks that have multiple ways of decomposition. Existing auction algo- rithmsforallocatingsuchcomplextaskscanbecategorizedintoeitherdecompose-then- allocate [GCD + 03, JBD + 06, Ort05] or allocate-then-decompose [BA99, MK01]. Both approaches suffer from decoupling the optimization problem into multiple stages and ignoring the inherent dependencies between them. Task tree auctions first model each complex task into a set of task trees, each of which represents one way of decomposi- tion of the complex task. Then, task tree auctions make robots evaluate and bid on task treesinsteadofatomictasksandthusallowrobotstobetterexpresstheirpreferenceson different ways of decomposition of a complex task. Finally, the auctioneer determines the set of nodes satisfying the task tree with the lowest team cost. Gerkey and Mataric [GM02]studyhowtoapplyMURDOCHtoatightly-coupledtaskdomain,namelybox pushing, where two robots act as pushers to cooperatively move a box from some ini- tial location to some goal location and one robot acts as watcher to monitor the current position of the box. The auction algorithm in MURDOCH makes the watcher be the auctioneer who keeps announcing and allocating push-box tasks of moving the box to certainpointstopushers. Robotsinthisalgorithmcanbeallocatedatmostonetaskand thustheydonotneedtosolveanyschedulingproblem. Thisalgorithmprovidestheflex- ibility of dynamically re-allocating tasks when some robot fails. For example, if only one pusher is available, the watcher still tries to allocate two push-box tasks as usual, butonlyonewillbeclaimedbytheremainingpusher. Afterthepusherfinishesthefirst push-box task, it will switch sides to execute the second one. Shehory and S. Kraus [SK95, SK98] develop greedy auction algorithms to address complex-task allocation 54 problems via coalition formations. They study two cases of the problem with disjoint andoverlappingcoalitions(thatis,whetherarobotcanexecutemultiplecomplextasks). Theyshowthattheiralgorithmsproduceasolutionthatiswithinalogarithmicfactorof theoptimal. However,theyassumethatthecostofrobotsexecutinganynewtaskisnot related to their allocated tasks. In this dissertation, we study complex tasks that require several robots to execute at the same time and there are positive or negative synergies amongtasks. NegotiationAlgorithms The first negotiation algorithm based on the interaction framework of the original con- tract net protocol is developed by M. Golfarelli et al. [GMR97a, GMR97b] for multi- robot path planning. After planning an initial path to execute its tasks, each robot uses thenegotiationprotocoltodecreaseitspathcostbyswappingtaskswiththeotherrobots. The negotiation protocol consists of three stages: i) robots first announce their current allocated tasks to each other; ii) robots then evaluate and make proposals of swapping single tasks with each other; and iii) a central planner finally collects all proposals and choosesthetaskswapthatreducesthesumofthepathcostsofallrobotsthemost. This negotiationprotocolisverysimilartotheone-to-onenegotiationprotocolshowninFig- ure 1.8 except that there does not exist a central planner in the one-to-one negotiation protocol. Actually,wecanimplementthisnegotiationalgorithmintheone-to-onenego- tiation protocol equivalently by making robots broadcast their best proposals to each other at the end of the proposal-evaluation phase and then locally figure out the task swap that can reduce the team cost the most. This negotiation algorithm also achieves the myopic optimality performance principle on single tasks we introduced on page 27 of Chapter 1. Similarly, Botelho and Alami [BA99] develop the M+ system based on theoriginalcontractnetprotocolinwhichrobotsnegotiatewitheachothertoexchange 55 tasks. They study how the negotiation process can be integrated into the robot control system. They propose a robot internal state diagram to describe the activities of robots forbothcooperativeinteractionandlocalplanning. The majority of the existing negotiation algorithms in the literature are based on the one-to-one negotiation protocol, namely, the resulting negotiation contract in each round involves only two robots. Lagoudakis et al. [LBK + 04] make use of single-item transfers between robots to improve the solution of sequential single-item auctions as follows: single tasks are transferred from one robot to another, starting with the task transfer that decreases the team cost the most, until no task transfer decreases the team cost any longer. M. Dias and A. Stentz [DS00] make use of single task swaps to allow robots to optimize their initial allocations as follows: each robot in turn has the chance tonegotiatewithalltheotherrobotstofindandimplementtheprofitabletaskswapsthat canreducetheirbothrobotcosts,untilnoprofitabletaskswapscanbefoundforanypair of robots. Sujit et al. [SSG06] study how to use a negotiation algorithm to coordinate multiple UAVs for searching and attacking targets on a battle field. UAVs have limited range of detecting targets and communicating with the other UAVs. Once a target is found by a UAV, it communicates with their neighbouring UAVs to determine whether it should execute the attack task by itself or transfer this attack task to some other UAV in its communication range. Sung et al. [SAR13] study task queue switching between robotsthateachneedtoexecuteatmostonequeueoftasks. Robotshavenoknowledge of tasks that are not in their queues but may share information about new tasks when theyareinthecommunicationrange. Robotsthennegotiatewitheachothertoexchange their task queues to improve the performance of the robot system when there are new tasks. Severalresearchersproposetousecombinatorialexchanges[PKE01]toconstructall possible negotiation contracts for robots to re-allocate their tasks. Sandholm and Suri 56 [SS00] show how to use combinatorial exchanges efficiently for E-commerce applica- tions. Dias and Stentz [DS02] use a leader in the robot system to run combinatorial exchanges to optimize the allocations for a subgroup of robots. Each robot in a combi- natorial exchange can perform an arbitrary number of task exchanges with an arbitrary number of robots at a time. Although combinatorial exchanges promise to always find theoptimalallocationsforrobots,itisknownthatsolvingtheclearingproblemofcom- binatorialexchangesisNP-hard[PKE01]. Researchwork[KSS04,SSS02]existsinthe literature that aims to make solving the clearing problem faster, however, they are all centralized algorithms and the robot systems still need to have a leader in these algo- rithms[SS00,DS02]. Sandholm [San98] study theoretical properties of negotiation contracts among robots for re-allocating tasks. He demonstrates that robots need to repeatedly perform theprofitableclustercontracts(thattransfermultipletasksfromonerobottoanotherata time)andmulti-robotcontracts(thattransfermultipletasksamongmorethantworobots at a time) in addition to the swap contracts that are commonly used in the literature in order to minimize the team cost. Lemaire et al. [LAL04] provide empirical results of allowingrobotstore-allocatebundlesofatmosttwotasksamongthem. Theystudyhow to balance the allocations of robots so that the task-completion time is as small as pos- sible. They introduce the concept of equity coefficient that characterizes the difference between the workload of a robot and the average workload of all robots. Each robot broadcastsitsownworkloadtotheotherrobotsandthencalculatesitsequitycoefficient independently. Tasksarere-allocatedamongrobotsinordertomaketheirequitycoeffi- cientssmall. ZhangandCollins[ZCB10,ZC12,ZCB13]developnegotiationalgorithms with stochastic clustering techniques. In their algorithms, robots randomly cluster their allocatedtasksandthenexchangeclustersoftasksamongeachother. Anovelfeatureof 57 their algorithms is that robots consider not only downhill movements (that is, perform- ing task exchanges that can decrease the team cost), but also uphill movements (that is, performing task exchanges that can increase the team cost), to avoid getting stuck in local minima during the negotiation process. They show that the team cost of the resulting allocations can be significantly reduced by turning the uphill movements on andoffintheiralgorithms. Cuietal. [CGG13]developnegotiationalgorithmsthatcan re-allocate multiple tasks among at most three robots at a time. Their algorithms eval- uate all possible task exchanges among three robots and perform the one that does not increaseanyrobotcostandreducesthesumoftherobotcostsofallrobotsthemost. 58 Chapter3 Simple-TaskAuctions Inthischapter,westudytask-allocationalgorithmsthatimplementtheiterativeauction protocol shown in Figure 1.5 for solving simple-task allocation problems. In a simple- taskallocationproblem,arobotcanfreelydeterminewhentoexecuteitsallocatedtasks and does not need to coordinate with other robots for task execution. At first glance, it seems to be easy to allocate such tasks to robots. In the literature, a significant num- berofauctionalgorithmshavebeenproposedandimplementedforsolvingsimple-task allocation problems, such as TraderBots [DZZS04] (CMU), CBAA [CBH09] (MIT), COMSTAR [HDPO07] (University of Nebraska), MAGNET [NG06] (University of Minnesota), MURDOCH [Ger03] (USC), and DEMiR-CF [Sar07] (Georgia Institute of Technology). However, it turns out that it is non-trivial to allocate even simple tasks to robots well because there exist synergies among tasks. Two tasks are said to have a positive (negative) synergy for a robot if the cost of the robot to execute both tasks together is smaller (larger) than the sum of the costs of executing them individually. A simple example is illustrative for this purpose. Figure 3.1 shows a multi-robot routing example with simple targets for the MiniMax team objective. Table 3.1 tabulates the robotcost(thesmallesttraveltime)ofeachrobotforvisitinganybundle(set)oftargets in this example. From this table, we can see that targets x 1 and x 2 exhibit a positive synergy for robot r 2 since the cost of robot r 2 for visiting both targets is 4, which is less than 6, the sum of the costs of robotr 2 for visiting each individual target. We can also see that targets x 1 and x 3 exhibit a negative synergy for robot r 2 since its cost for visitingbothtargetsis12,muchlargerthan8,thesumoftheindividualcosts. 59 Solvingtask-allocationproblemswithsynergiesamongtasksoptimallyisknownto be NP-hard for all three team objectives [Ger03, TLJK05, RKPT03]. Thus, one cannot expect to develop task-allocation algorithms that always result in optimal allocations withinarealisticresponsetime. Manyexistingauctionalgorithmsusesequentialsingle- item auctions to allocate tasks to robots. In a sequential single-item auction, tasks are allocated to robots in multiple rounds and one additional task is allocated per round so that the team cost increases the least. Although sequential single-item auctions can allocatetasksquickly,theauctioneerconsidersverylimitedsynergiesamongtaskswhen it makes allocation decisions and thus sequential single-item auctions usually result in highlysub-optimalallocations. In this chapter, we illustrate that sequential single-item auctions can be extended to considermoresynergiesamongtasksandresultinbetterallocationsbyallocatingmore than one task to robots in each round. Our goal is to develop auction algorithms for solving simple-task allocation problems (simple-task auctions) that achieve the follow- ingmyopicoptimizationperformanceprincipleonmultipletasks: Performanceprincipleofsimple-taskauctions: Toallocateagivennum- berofunallocatedtaskstooneormorerobotsineachroundofsimple-task auctionssothatthecurrentteamcostincreasestheleast. Weproceedasfollows: Section3.1formalizessimple-taskallocationproblems. Sec- tion3.2summarizesseveralexistingsimple-taskauctionalgorithms. InSection3.3,we first introduce a novel data structure, called bid trees, that succinctly characterizes the local information of robots necessary to achieve the performance principle of simple- task auctions. After that, we develop a new auction algorithm, called sequential auc- tions with bundles, in which robots generate and submit bids from their bid trees to the auctioneerandtheauctioneerallocatesagivennumberofunallocatedtaskstorobotsin 60 r 1 x 1 x 2 r 2 x 3 Figure3.1: Example4ofMulti-RobotRouting {x 1 } {x 2 } {x 3 } {x 1 ,x 2 } {x 1 ,x 3 } {x 2 ,x 3 } {x 1 ,x 2 ,x 3 } r 1 4 6 12 6 12 12 12 r 2 4 2 4 4 12 8 12 Table3.1: RobotCostsforExampleinFigure3.1(MiniMaxTeamObjective) eachroundsothatthecurrentteamcostincreasestheleast. Finally,Section3.4presents experimentalresults,andSection3.5concludesthischapter. 3.1 ProblemDescription We now formalize simple-task allocation problems. A simple-task allocation problem ⟨R,X,E⟩ consists of a finite set of robots R = {r 1 ,...,r m }, a finite set of tasks X ={x 1 ,...,x n }, and a finite set of capabilitiesE ={e 1 ,...,e o }. Each robotr∈ R ischaracterizedbyonecapabilitye r ∈E thatitprovides. Eachtaskx∈X ischaracter- izedbyonecapabilitye x ∈E thatitrequires. Arobotr issaidtobequalifiedforatask x iffe r = e x . Each task needs to be executed by exactly one robot that is qualified for it. Eachrobotr∈Risalsocharacterizedbyataskcapacityq r ,withthemeaningthatit canexecuteatmostq r tasks. Any tuple (X r 1 ,...,X rm ) of pairwise disjunct bundles of tasks X r i ⊆ X, for all i = 1...m,isapartialsolutionofthesimple-taskallocationproblem,withthemeaning thatrobotr i isallocatedtoexecutethetasksinX r i . 61 For any given execution order of tasks in X ′ that are allocated to robot r, the task- completiontimeofrobotristhelargestexecutiontimeofanytask(thatis,theexecution timeofthelasttask)inthisexecutionorderandthe cumulative latencyofrobotr isthe sum of execution times of all tasks in this execution order. The definition of robot cost c robot r (X ′ ) of robot r to execute tasks X ′ depends on the given team objective. For both MiniSum and MiniMax team objectives, the robot cost is the smallest task- completiontimeofrobotr forexecutingtasksX ′ inanoptimalexecutionorder(among all possible execution orders). For the MiniLat team objective, the robot cost is the smallest cumulative latency of robot r for executing tasks X ′ in an optimal execution order. We define the robot costc robot r (X ′ ) =∞ if|X ′ | > q r or robotr is not qualified for one or more tasks in X ′ . There can be synergies among tasks for robot r, that is, c robot r (X ′ )+c robot r (X ′′ )doesnotnecessarilyequalc robot r (X ′ ∪X ′′ )evenifX ′ ∩X ′′ =∅ andc robot r (X ′ ∪X ′′ )<∞. The definition of the team cost of any partial solution (X r 1 ,...,X rm ) also depends on the given team objective. For the MiniSum and MiniLat team objectives, the team cost of the partial solution is ∑ r∈R c robot r (X r ). For the MiniMax team objective, the team cost is max r∈R c robot r (X r ). In this dissertation, we use c team as a special operator (shorthand) for either the sum and max operator, depending on the team objective, and writec team r∈R c robot r (X r )tomakeournotationindependentoftheteamobjective. Any partial solution (X r 1 ,...,X rm ) with ∪ r∈R X r = X (each task is allocated to exactly one robot) is a complete solution of the simple-task allocation problem. We wanttofindacompletesolutionofthesimple-taskallocationproblemwithasmallteam cost 1 . 1 We assume that every simple-task allocation problem has at least one complete solution in this dis- sertation. 62 3.2 ExistingApproachestoSimple-TaskAuctions In this section, we discuss several state-of-the-art auction algorithms in the literature thatsolvesimple-taskallocationproblems,namely,combinatorialauctions,parallelauc- tions,andsequentialsingle-itemauctions. 3.2.1 CombinatorialAuctions A combinatorial auction [CSS06] consists of exactly one round to allocate tasks to robots. Initially, all tasks are unallocated. During this round, all robots bid on every non-emptybundleofunallocatedtasksandthentheauctioneerallocatesallunallocated tasks to robots so that the team cost increases the least. Assume the current allocation of each robot is∅ and U = X is the set of unallocated tasks. We now explain exactly whichbidsrobotssubmitandhowtheauctioneermakestheallocationdecision. • Bid-ConstructionPhase: Arobotr∈RiseligibleforabundleoftasksX ′ ⊆X iff|X ′ |≤q r androbotr isqualifiedforalltasksx∈X ′ . Forallnon-emptybundlesofunallocatedtasksX ′ ⊆U,ifrobotr∈Riseligible forX ′ ,thenitcalculatesitsrobotcostc robot r (X ′ )forexecutingalltasksinX ′ and thensubmitstherobotcostc robot r (X ′ )asitsbidonbundleX ′ totheauctioneer. • Winner-DeterminationPhase: Theauctioneerfirstconstructsallpossible(com- plete)solutions(X r 1 ,...,X rm ). Afterthat,theauctioneercalculatestheresulting teamcostc team r∈R c robot r (X r )foreachsolution (X r 1 ,...,X rm ),wheretherobotcost c robot r (X r )is0ifX r =∅orthebidsubmittedbyrobotr onbundleX r inthebid constructionphaseotherwise. Finally,theauctioneerdeterminestheoptimalallo- cations(X r 1 ,...,X rm )withthesmallestteamcost(tiescanbebrokenarbitrarily) andallocatesthetasksinbundleX r torobotr foreachrobotr∈R. 63 BidConstruction WinnerDetermination Bids(calculatedandsubmitted) Allocations(evaluated) Decision Round1 Robotr 1 : 4onbundle{x 1 } ({x 1 ,x 2 ,x 3 },∅): 12 6onbundle{x 2 } ({x 1 ,x 2 },{x 3 }): 6 12onbundle{x 3 } ({x 1 ,x 3 },{x 2 }): 12 Theauctioneer 6onbundle{x 1 ,x 2 } ({x 2 ,x 3 },{x 1 }): 12 allocates{x 1 ,x 2 } 12onbundle{x 1 ,x 3 } ({x 1 },{x 2 ,x 3 }): 8 torobotr 1 and 12onbundle{x 2 ,x 3 } ({x 2 },{x 1 ,x 3 }): 12 {x 3 }torobotr 2 12onbundle{x 1 ,x 2 ,x 3 } ({x 3 },{x 1 ,x 2 }): 12 (∅,{x 1 ,x 2 ,x 3 }): 12 Robotr 2 : 4onbundle{x 1 } 2onbundle{x 2 } 4onbundle{x 3 } 4onbundle{x 1 ,x 2 } 12onbundle{x 1 ,x 3 } 8onbundle{x 2 ,x 3 } 12onbundle{x 1 ,x 2 ,x 3 } Table 3.2: Combinatorial Auctions for Example in Figure 3.1 (MiniMax Team Objec- tive) r 1 x 1 x 2 r 2 x 3 OptimalAllocations AllocationsofCombinationalAuctions (TeamCost=6) r 1 x 1 x 2 r 2 x 3 AllocationsofParallelAuctions AllocationsofSequentialSingle-ItemAuctions AllocationsofSequentialAuctionswithBundleSizeTwo (TeamCost=12) Figure3.2: AllocationsforExampleinFigure3.1(MiniMaxTeamObjective) Consider again the multi-robot routing example shown in Figure 3.1 for the Mini- Max team objective. There exists only one capability. Robots r 1 and r 2 provide this capability,andtargetsx 1 ,x 2 andx 3 requireit. Thecapacityofbothrobotsis3,andthus they are eligible for all possible bundles of targets in this example. Table 3.2 tabulates the bids calculated and submitted by robots in the bid construction phase of combi- natorial auctions. Table 3.2 also tabulates all possible (complete) solutions (with the corresponding team costs) the auctioneer has to construct and evaluate in the winner determination phase. Finally, the auctioneer determines to allocate targets{x 1 ,x 2 } to 64 robotr 1 and targetx 3 to robotr 2 since the team cost 6 (shown in bold in Table 3.2) of the solution (X r 1 ={x 1 ,x 2 },X r 2 ={x 3 }) is smallest among all possible ones. The left part of Figure 3.2 shows the resulting allocations of combinatorial auctions for the MiniMaxteamobjective. The team cost of combinatorial auctions is smallest since all positive and negative synergies among tasks have been taken into account by both bidders and the auction- eer. However, there are several limitations for applying them in practice: i) robots have to evaluate a large number of bundles of tasks (that is exponential in the number of tasks), which is not computation efficient; ii) robots have to submit a a large num- ber of bids (that is exponential in the number of tasks) to the auctioneer, which is not communication efficient; and iii) solving the winner-determination problem of the auc- tioneeroptimallyincombinatorialauctionsisNP-hard[HB00],andthustheauctioneer becomes the bottleneck of the robot system even though there exist some approaches [Kwo02,MV07]intheliteraturethatspeedupthewinner-determinationprocess. 3.2.2 ParallelAuctions A parallel auction [DBBB08] consists of exactly one round to allocate tasks to robots. Initially, all tasks are unallocated. During this round, all robots bid on each single unallocated task and then the auctioneer allocates all unallocated tasks to robots so that eachtaskisallocatedtotherobotthatsubmitsthesmallestbidonit(tiescanbebroken arbitrarily). Assume the current allocation of each robot is∅, and U = X is the set of unallocated tasks. We now explain exactly which bids robots submit and how the auctioneermakestheallocationdecision. • Bid-Construction Phase: For each unallocated task x∈ U, if robot r∈ R is eligible for executing it, then it calculates its robot costc robot r ({x}) for executing 65 BidConstruction WinnerDetermination Bids(calculatedandsubmitted) Allocations(evaluated) Decision Round1 Robotr 1 : 4ontargetx 1 (x 1 →r 1 ): 4 6ontargetx 2 (x 1 →r 2 ): 4 12ontargetx 3 Theauctioneerallocates (x 2 →r 1 ): 6 targetx 2 torobotr 2 , Robotr 2 : 4ontargetx 1 (x 2 →r 2 ): 2 targetx 1 torobotr 2 (bytiebreaking), 2ontargetx 2 andtargetx 3 torobotr 2 4ontargetx 3 (x 3 →r 1 ): 12 (x 3 →r 2 ): 4 Table3.3: ParallelAuctionsforExampleinFigure3.1(MiniMaxTeamObjective) the single task x and then submits this robot cost c robot r ({x}) as its bid on task x totheauctioneer. • Winner-DeterminationPhase: LetB be the set of all bids submitted by robots. The auctioneer allocates tasks to robots by the following steps: It first finds the smallest bid in set B (ties can be broken arbitrarily). Assume that the selected winning bid is submitted by robotr on taskx. It then allocates taskx to robotr. After that, it removes the bids from set B if i) that are on task x, or ii) that are submittedbyrobotr androbotr isnolongereligibleforexecutinganyadditional task. ItrepeatsthisprocedureuntilsetB isempty. Consider again the multi-robot routing example shown in Figure 3.1 for the Mini- Max team objective. Table 3.3 tabulates the bids calculated and submitted by robots in parallel auctions. Each robot submits bids only on single targets instead of all possible bundles of targets. Table 3.3 also tabulates the single-task allocations the auctioneer needs to evaluate. Since each robot is eligible for visiting all targets in this example, the auctioneer allocates each target to the robot who submits the smallest bid on it (ties are broken arbitrarily). The right part of Figure 3.2 shows the resulting allocations of parallelauctionsfortheMiniMaxteamobjective. Parallel auctions are easy to understand and simple to implement. The computation and communication of both robots and the auctioneer are efficient since: i) the number of bids each robot needs to evaluate and submit is at most the number of tasks in the 66 bid-constructionphase,andii)thetotalnumberofbidstheauctioneerneedstoevaluate is at most the number of robots times the number of tasks in the winner-determination phase. However, no synergies among tasks are taken in account in parallel auctions since each robot bids on only single tasks. The team cost of parallel auctions for the MiniSum team objective can be arbitrarily far away from the smallest when robots are eligibleforexecutingalltasks[LBK + 04]. 3.2.3 SequentialSingle-ItemAuctions Nowwediscusssequentialsingle-itemauctions[TLJK05],whichcombinesomeadvan- tages of combinatorial auctions and parallel auctions. A sequential single-item auction consistsofmultipleroundstoallocatetaskstorobots. Initially,alltasksareunallocated. During each round, each robot bids on each single unallocated task and then the auc- tioneer allocates one additional unallocated task to some robot so that the current team costincreasestheleast. Wenowconsideranyroundofasequentialsingle-itemauction. Assumethateachrobotr∈RhasbeenallocatedthetasksX r ⊆X inpreviousrounds. Thus,U = X\ ∪ r∈R X r isthesetofunallocatedtasks. Wenowexplainexactlywhich bidsrobotssubmitandhowtheauctioneermakestheallocationdecision. • Bid-Construction Phase: For each unallocated task x ∈ U, if robot r ∈ R is eligible for tasks X r ∪{x}, then it calculates its bid on task x as follows: i) c robot r (X r ∪{x})−c robot r (X r ) for the MiniSum and MiniLat team objectives, namely,robotr bidstheincreaseinitsrobotcostforexecutingalltasksallocated to it if it is additionally allocated task x, which is similar to previous work on marginal-cost bidding in TRACONET [San96]; and ii) c robot r (X r ∪{x}) for the MiniMax team objective, namely, robot r bids its robot cost for executing all tasks allocated to it if it is additionally allocated task x. After that, robot r finds 67 BidConstruction WinnerDetermination Bids(calculated) Bids(submitted) Allocations(evaluated) Decision Round1 Robotr 1 : 4ontargetx 1 6ontargetx 2 Robotr 1 bids4ontargetx 1 (x 1 →r 1 ): 4 12ontargetx 3 (x 2 →r 2 ): 2 Theauctioneer allocatestarget Robotr 2 : 4ontargetx 1 x 2 torobotr 2 2ontargetx 2 Robotr 2 bids2ontargetx 2 4ontargetx 3 Round2 Robotr 1 : 4ontargetx 1 Robotr 1 bids4ontargetx 1 12ontargetx 3 (x 1 →r 1 ): 4 Theauctioneer (x 1 →r 2 ): 4 allocatestarget Robotr 2 : 4ontargetx 1 Robotr 2 bids4ontargetx 1 x 1 torobotr 2 8ontargetx 3 (bytiebreaking) Round3 Robotr 1 : 12ontargetx 3 Robotr 1 bids12ontargetx 3 (x 3 →r 1 ): 12 Theauctioneer (x 3 →r 2 ): 12 allocatestarget Robotr 2 : 12ontargetx 3 Robotr 2 bids12ontargetx 3 x 3 torobotr 2 (bytiebreaking) Table 3.4: Sequential Single-Item Auctions for Example in Figure 3.1 (MiniMax Team Objective) itssmallestbidoveralltasksinU (tiescanbebrokenarbitrarily)andthensubmits onlythatbidtotheauctioneer. • Winner-DeterminationPhase: LetB besetofallbidssubmittedbyrobots. The auctioneer first finds the smallest bid in set B (ties can be broken arbitrarily). Assume that the selected winning bid is submitted by robot r on task x. It then allocatestaskxtorobotr. Consider again the multi-robot routing example shown in Figure 3.1 for the Mini- Max team objective. Table 3.4 shows the allocation process of sequential single-item auctions for this example. It consists of three rounds to allocate three targets to robots. One additional target is allocated to one robot in each round. Table 3.4 tabulates the bids calculated and submitted by robots in each round of sequential single-item auc- tions. (Robots do not need to visit targets in the order in which targets are allocated to themwhentheycalculatetheirrobotcosts.) Differentfromcombinationalauctionsand parallel auctions, the bids calculated and submitted by robots in sequential single-item auctions are no longer identical: a robot calculates its bids for each unallocated target, butitsubmitsonlythesmallestone. Thereasonisthatarobotcanlocallyinferthatonly 68 itssmallestbidhasachancetowinsincethewinningbiddeterminedbytheauctioneer isthesmallestoverallbidsonsingletasks(tiesarebrokenbytheauctioneerinfavorof submitted bids). Table 3.4 also tabulates the possible allocations the auctioneer needs to evaluate and its allocation decision in each round of sequential single-item auctions. The right part of Figure 3.2 shows the resulting allocations of sequential single-item auctionsfortheMiniMaxteamobjective. Sequential single-item auctions are easy to understand and simple to implement. The computation and communication of both robots and the auctioneer are efficient 2 since: i)thenumberofbidseachrobotneedstoevaluateislinearinthenumberoftasks, and each robot needs to submit only a single bid to the auctioneer per round in the bid- constructionphase;andii)thetotalnumberofbidstheauctioneerneedstoevaluateper round is linear in the number of robots and independent of the number of tasks in the winner-determinationphase. Sequentialsingle-itemauctionscantakeintoaccountsomebuttypicallynotallsyn- ergiesamongtasks. Theytakeintoaccountthesynergiesbetweeneachunallocatedtask andthetasksthathavealreadybeenallocatedtorobotsinpreviousroundswhenrobots calculate bids on the additional unallocated single tasks. For example, Table 3.4 shows thatrobotr 2 calculatesitsbid 8ontargetx 3 inRound2whilethesumofitsrobotcosts of visiting targets x 3 and x 2 individually is 6, which illustrates that robot r 2 does take into account the negative synergy between the unallocated target x 3 and its allocated targetx 2 . Sequential single-item auctions achieve the myopic optimization performance prin- ciple on single tasks, namely, to allocate one additional unallocated task to some robot 2 We do not take into account the computational efforts of robot r needs to calculate its robot cost c robot r (X r )ofexecutingtasksX r ,whichisdomain-dependent. Forexample,robotsinmulti-robotrouting needtosolveNP-hardtravelingsalespersonproblems(exceptthattheydonotneedtoreturntotheirinitial locations) for calculating their robot costs and thus need to use approximation algorithms to speed up in ourexperiments. 69 Sequential Single-Item Auctions Tasks Allocated per Round Synergies Considered ∞ 1 all limited Unexplored Area Combinatorial Auctions Figure 3.3: Connection between Combinatorial Auctions and Sequential Single-Item Auctions per round so that the team cost increases the least [TLJK05]. However, the team cost of the solution resulting from sequential single-item auctions can be still highly sub- optimal,giventhatonlysomebuttypicallynotallsynergiesamongtasksaretakeninto account. 3.3 SequentialAuctionswithBundles In this section, we develop a new auction algorithm that can improve the team perfor- manceofsequentialsingle-itemauctionsandmaintaintheirsimplicityatthesametime. Consider again combinatorial auctions and sequential single-item auctions in terms of the synergies among tasks that are taken into account by robots and the auctioneer. In a combinatorial auction, robots take into account all positive and negative syner- gies among tasks since they evaluate all possible non-empty bundles of tasks in the bid-construction phase, and the auctioneer allocates all tasks to robots in one round to minimize the resulting team cost in the winner-determination phase. Thus, the team performance of combinatorial auctions is optimal. In a sequential single-item auction, robots take into account some but typically not all synergies among tasks since they 70 evaluate only single unallocated tasks in the bid-construction phase, and the auctioneer allocatesonlyoneadditionaltasktosomerobotsothattheteamcostincreasestheleast in the winner-determination phase. Thus, the team performance of sequential single- itemauctionsisnotguaranteedtobeoptimal. We want to improve the team performance of sequential single-item auctions by exploringthesimilarityandthedifferencebetweencombinatorialauctionsandsequen- tialsingle-itemauctions. Onecanviewsingletasksasbundlesofonetask. Thus,robots in sequential single-item auctions evaluate bundles of one task and in combinatorial auctionsevaluateallpossiblenon-emptybundlesoftasksinthebid-constructionphase. The main difference between combinatorial auctions and sequential single-item auc- tions in the bid-construction phase is the size of bundles of tasks evaluated by robots. Similar reasoning also applies for the auctioneer in the winner-determination phase. In sequential single-item auctions, the auctioneer allocates one additional unallocated task to some robot per round so that the team cost increases the least. In combinato- rial auctions, the auctioneer allocates all unallocated tasks to robots per round so that the team cost increases the least. The main difference between combinatorial auctions and sequential single-item auctions in the winner-determination phase is the number of unallocatedtasksallocatedtorobotsperround. Figure3.3demonstratesthisconnectionbetweensequentialsingle-itemauctionsand combinatorial auctions. Sequential single-item auctions and combinatorial auctions are two extreme cases in terms of the size of bundles of tasks evaluated by robots and the number of tasks allocated by the auctioneer per round. There still exists a large unex- ploredareabetweenthesetwoapproaches. Thus,itisnaturaltoconjecturethatincreas- ing the size of bundles of tasks evaluated by robots and the number of tasks allocated bytheauctioneerperroundcantakeintoaccountmoresynergiesamongtasksandthus improvetheteamperformanceofsequentialsingle-itemauctions. 71 Sequential Single-Item Auctions Combinatorial Auctions Tasks Allocated per Round Synergies Considered k = ∞ all limited Sequential Auctions with Bundle Size k k k = 1 the more the larger k is Figure 3.4: Connection between Sequential Auctions with Bundles and Existing Auc- tionAlgorithms In this section, we develop a new auction algorithm for solving simple-task alloca- tion problems, called sequential auctions with bundles, in which each robot considers a larger size of bundles of tasks than sequential single-item auctions and the auctioneer allocatesmorethanonetasktorobotssothatthecurrentteamcostincreasestheleastin eachround. AsshowninFigure3.4,sequentialauctionswithbundlesexploretheempty areabetweencombinationalauctionsandsequentialsingle-itemauctions. 3.3.1 Overview Sequential auctions with bundles consist of multiple rounds to allocate tasks to robots, which is similar to sequential single-item auctions. Initially, all tasks are unallocated. During each round, all robots bid on all non-empty bundles of at most k unallocated tasksinthebid-constructionphaseandtheauctioneerthenallocatesexactlykadditional unallocatedtaskstorobots(oralltasksincasethenumberofunallocatedtasksissmaller thank), either to the same robot or different robots, in the winner-determination phase. Wecallk thebundlesizeofthesequentialauctionwithbundles. 72 Wenowconsideranyroundofasequentialauctionwithbundlesizek. Assumethat each robot r∈ R has already been allocated the tasks X r ⊆ X in previous rounds. Thus,U =X\ ∪ r∈R X r isthesetofunallocatedtasks. Weleaveoutthe“unallocated” inthefollowingforreadabilitysinceonlyunallocatedtasksarebidonandsubsequently allocated. A bidb in sequential auctions with bundles is a triple of a robot, a bundle of tasks and a bid cost (numerical value of the bid). Ifb is a bid, then we useb r to denote the robot, b x to denote the bundle the robot bids on and b c to denote the bid cost. We now explain exactly which bids robots submit (including how much they bid) and how theauctioneermakestheallocationdecision. • Bid-ConstructionPhase: Arobotr∈RiseligibleforabundleoftasksX ′ ⊆X iff|X r ∪X ′ |≤q r androbotr isqualifiedforalltasksx∈X ′ . ForeachbundleoftasksX ′ ⊆ U with 0 <|X ′ |≤ k, ifrobotr iseligibleforX ′ , thenitconstructsandsubmitsthefollowingbidbonbundleX ′ totheauctioneer: i)b r =r;ii)b x =X ′ andiii)b c =c robot r (X r ∪X ′ )−c robot r (X r )fortheMiniSum andMiniLatteamobjectives,namely,robotrbidstheincreaseinitsrobotcostfor executing all tasks allocated to it if it is additionally allocated the tasks in bundle X ′ , which is similar to previous work on marginal-cost bidding in TRACONET [San96];orb c =c robot r (X r ∪X ′ )fortheMiniMaxteamobjective,namelyrobotr bidsitsrobotcostforexecutingalltasksallocatedtoitifitisadditionallyallocated thetasksinbundleX ′ . • WinnerDeterminationPhase: LetB be the set of all bids submitted by robots. EachcollectionB ′ ⊆Bofbidsispotentiallywinningiff1)b r ̸=b ′ r andb x ∩b ′ x =∅ for all b,b ′ ∈ B ′ with b̸= b ′ (a potentially winning collection of bids must have beenmadebydifferentrobotssinceseveralbidsbythesamerobotdonotexpress synergies); and 2)| ∪ b∈B ′ b x | = min(k,|U|) (a potentially winning collection 73 of bids must cover exactly k tasks or all tasks in case the number of tasks is smaller than k). We denote the set of all potentially winning collections of bids as P(B). The auctioneer evaluates a potentially winning collection B ′ ∈ P(B) of bids according to the value c team b∈B ′b c , called evaluation cost, for the given team objective. Thus, the winning collection B w ∈ P(B) of bids satisfies, for all B ′ ∈ P(B), c team b∈Bw b c ≤ c team b∈B ′b c (the winning collection of bids must have the smallest evaluation cost among all potentially winning collections of bids). Ties can be broken arbitrarily. The auctioneer then allocates the tasks b x to robot b r for all b∈ B w . At this point in time, robot r∈ R has been allocated the tasks X r (B w ) = X r ∪ b x if there exists b∈ B w with b r = r, and X r (B w ) = X r otherwise. We now explain why we expect these bidding and winner-determination rules to resultinasmallteamcost. Theorem3.3.1 For all B ′ ∈ P(B), it holds that c team r∈R c robot r (X r (B w )) ≤ c team r∈R c robot r (X r (B ′ )). Proof: Let Y and Z be any two potentially winning collections of bids. We want to show that c team b∈Y b c ≤ c team b∈Z b c implies c team r∈R c robot r (X r (Y)) ≤ c team r∈R c robot r (X r (Z)). For any potentially winning collections of bids B, let R(B) be the set of robots b r for each bid b ∈ B, and B r be the (possibly empty) bundle of tasks bid on by robot r in B. Consider the MiniSum and MiniLat team objectives. By def- inition, c team r∈R c robot r (X r (Y)) = ∑ r∈R c robot r (X r (Y)) = ∑ r∈R c robot r (X r ∪ Y r ) = ∑ r∈R(Y) c robot r (X r ∪ Y r ) + ∑ r∈R\R(Y) c robot r (X r ). For each b ∈ Y, we have b c = c robot br (X br ∪Y br )−c robot br (X br ). Thus,c team r∈R c robot r (X r (Y)) =c team b∈Y b c +c team r∈R (X r ). Sim- ilarly, we have c team r∈R c robot r (X r (Z)) = c team b∈Z b c + c team r∈R (X r ). Thus, c team b∈Y b c ≤ c team b∈Z b c impliesc team r∈R c robot r (X r (Y))≤c team r∈R c robot r (X r (Z)). 74 BidConstruction WinnerDetermination Bids(calculated) Bids(submitted) Allocations(evaluated) Decision Round1 Robotr 1 : 4onbundle{x 1 } Robotr 1 : 4onbundle{x 1 } ({x 1 ,x 2 }→r 1 ): 6 6onbundle{x 2 } 6onbundle{x 2 } ({x 1 ,x 3 }→r 1 ): 12 12onbundle{x 3 } 12onbundle{x 3 } ({x 2 ,x 3 }→r 1 ): 12 6onbundle{x 1 ,x 2 } 6onbundle{x 1 ,x 2 } ({x 1 ,x 2 }→r 2 ): 4 Theauctioneer 12onbundle{x 1 ,x 3 } 12onbundle{x 1 ,x 3 } ({x 1 ,x 3 }→r 2 ): 12 allocatestargets 12onbundle{x 2 ,x 3 } 12onbundle{x 2 ,x 3 } ({x 2 ,x 3 }→r 2 ): 8 x 1 andx 2 to robotr 2 (by Robotr 2 : 4onbundle{x 1 } Robotr 2 : 4onbundle{x 1 } (x 1 →r 1 ,x 2 →r 2 ): 4 tiebreaking) 2onbundle{x 2 } 2onbundle{x 2 } (x 1 →r 1 ,x 3 →r 2 ): 4 4onbundle{x 3 } 4onbundle{x 3 } (x 2 →r 1 ,x 1 →r 2 ): 6 4onbundle{x 1 ,x 2 } 4onbundle{x 1 ,x 2 } (x 2 →r 1 ,x 3 →r 2 ): 6 12onbundle{x 1 ,x 3 } 12onbundle{x 1 ,x 3 } (x 3 →r 1 ,x 1 →r 2 ): 12 8onbundle{x 2 ,x 3 } 8onbundle{x 2 ,x 3 } (x 3 →r 1 ,x 2 →r 2 ): 12 Round2 Robotr 1 : 12onbundle{x 3 } Robotr 1 : 12onbundle{x 3 } (x 3 →r 1 ): 12 Theauctioneer allocatestarget Robotr 2 : 12onbundle{x 3 } Robotr 2 : 12onbundle{x 3 } (x 3 →r 2 ): 12 x 3 torobotr 2 (bytiebreaking) Table3.5: SequentialAuctionswithBundleSizeTwoforExampleinFigure3.1(Mini- MaxTeamObjective) Now consider the MiniMax team objective. By definition, c team r∈R c robot r (X r (Y)) = max r∈R c robot r (X r (Y)) = max{max r∈R\R(Y) c robot r (X r ),max r∈R(Y) c robot r (X r ∪Y r )} = max{max r∈R c robot r (X r ),max r∈R(Y) c robot r (X r ∪Y r )}. For each b∈ Y, we have b c = c robot br (X br ∪ Y br ). Thus, c team r∈R c robot r (X r (Y)) = max{c team r∈R c robot r (X r ),c team b∈Y b c }. Sim- ilarly, c team r∈R c robot r (X r (Z)) = max{c team r∈R c robot r (X r ),c team b∈Z b c }. Thus, c team b∈Y b c ≤ c team b∈Z b c impliesc team r∈R c robot r (X r (Y))≤c team r∈R c robot r (X r (Z)). Theorem 3.3.1 says that each round of sequential auctions with bundles allocates min(k,|U|) additional tasks to robots so that the team cost after these allocations is as smallaspossible,wherekisagivenparameter. Inotherwords,sequentialauctionswith bundlesachievethemyopicoptimalityperformanceprincipleonmultipletasks,namely, toallocateagivennumberofadditionaltaskstoeitherthesamerobotordifferentrobots ineachroundsothatthecurrentteamcostincreasestheleast. Sequentialauctionswith bundlesalsogeneralizesequentialsingle-itemauctionswherethebundlesizeisoneand combinatorial auctions where the bundle size is the total number of tasks, as shown in Figure3.4. 75 Consider again the multi-robot routing example shown in Figure 3.1 for the Min- iMax team objective. Table 3.5 tabulates the bids calculated and submitted by robots andtheallocationstheauctioneerhastoevaluateinsequentialauctionswithbundlesize two for this example. It takes two rounds to allocate three targets to robots. In the first round, each robot calculates and submits bids on all non-empty bundles of at most two targets and the auctioneer evaluates all possible allocations of two targets to either the same robot or different robots. In the second round, since there is only one unallocated target left, sequential auctions with bundle size two behave in the same way as sequen- tial single-item auctions. The right part of Figure 3.2 shows the resulting allocations of sequentialauctionswithbundlesizetwofortheMiniMaxteamobjective. Although sequential auctions with a larger bundle size consider more synergies among tasks and are promising to improve the team performance of sequential single- item auctions, there are two technical difficulties for applying sequential auctions with bundles(asdescribedsofar): • Difficulty 1: The number of bids each robot needs to submit to the auctioneer is atleast ( n k ) (thenumberofpossiblebundlesofexactlyktasksfromntasks)where nisthetotalnumberoftasksandkisbundlesize,whichdoesnotscalewellwhen thenumberoftasksincreasesandmayresultinacommunicationbottleneck. • Difficulty2: Thenumberofallocationstheauctioneerneedstoevaluatebasedon the bids submitted from robots is at least m k ( n k ) (the number of possible alloca- tions of exactlyk tasks fromn tasks tom robots) wherem is the total number of robots, which does not scale well when the number of robots and tasks increases andmayresultinacomputationbottleneck. Given these two difficulties, sequential auctions with bundles do not scale well to largenumbersofrobotsandtaskswhenthebundlesizeislargerthan2. Inthefollowing 76 two subsections, we develop techniques for addressing these difficulties so that every robot needs to submit only a constant number of bids per round and the runtime of winner determination is linear in the number of robots and independent of the number of tasks (for a given bundle size). This is perhaps surprising since sequential auctions with bundles are similar to combinatorial auctions, for which winner determination is knowntobeNP-hard[HB00]. 3.3.2 Bid-ConstructionPhase Nowwestudywhichbundlesoftasksrobotsneedtosubmitbidstotheauctioneeronand which bundles they can safely ignore. Consider again sequential single-item auctions (sequential auctions with bundle size one). A robot in a sequential single-item auction calculates bids on all single tasks in the bid-construction phase, however, it does not submitallofthemtotheauctioneerbutonlytheonewiththesmallestbidcost(tiescan bebrokenarbitrarily). Thereasonisthatrobotscanlocallyinferthatotherbidshaveno chanceofwinningiftheauctioneerbreakstiesamongbidswithequalbidcostsinfavor ofthesubmittedones. Thesameideaalsoappliestosequentialauctionswithbundlesize two. Robots in a sequential auction with bundle size two (as described so far) calculate andsubmittheirbidsonallbundlesoftwotasksandonallbundlesofsingletasksinthe bid-construction phase. In the following, we show that robots do not need to submit all of them to the auctioneer. The auctioneer in a sequential auction with bundle size two allocates two tasks in the winner-determination phase. It should allocate the winning two tasks either to the same robot (Case 1) or to two different robots (Case 2). In Case 1, the winning bid must be submitted by a single robot on a bundle of two tasks. Thus, each robot can locally infer that only the bid with the smallest bid cost among all its bidsonbundlesoftwotaskshasachanceofwinning(theauctioneerbreakstiesinfavor of the submitted bids). As a result, each robot needs to submit only one bid on bundles 77 BidConstruction Bids(calculated) Bids(submitted) Round1 Robotr 1 : 4onbundle{x 1 } Robotr 1 : 4onbundle{x 1 } 6onbundle{x 2 } 6onbundle{x 2 } 12onbundle{x 3 } ———————– 12onbundle{x 3 } 6onbundle{x 1 ,x 2 } 6ontargets{x 1 ,x 2 } 12onbundle{x 1 ,x 3 } —————————- 12onbundle{x 1 ,x 3 } 12onbundle{x 2 ,x 3 } —————————- 12onbundle{x 2 ,x 3 } Robotr 2 : 4onbundle{x 1 } Robotr 2 : ——————– 4onbundle{x 1 } 2onbundle{x 2 } 2onbundle{x 2 } 4onbundle{x 3 } 4onbundle{x 3 } 4onbundle{x 1 ,x 2 } 4onbundle{x 1 ,x 2 } 12onbundle{x 1 ,x 3 } —————————- 12onbundle{x 1 ,x 3 } 8onbundle{x 2 ,x 3 } —————————- 8onbundle{x 2 ,x 3 } Table3.6: UnnecessaryBidsSubmittedbyRobotsinTable3.5 of two tasks to the auctioneer. In Case 2, the winning bids must be submitted by two differentrobotsontwodifferentsingletasks. Thus,eachrobotcanlocallyinferthatonly itstwobidswiththesmallestbidcosts 3 amongallitsbidsonsingletaskshaveachance ofwinning(theauctioneerbreakstiesinfavorofthesubmittedbids). 4 Asaresult,each robotneedstosubmitonlytwobidsonsingletaskstotheauctioneer. Puttogether,each robotinasequentialauctionwithbundlesizetwoneedstosubmitonlythreebidstothe auctioneerinthebid-constructionphase,namely,onebidonbundlesoftwotaskswhose bidcostissmallestandtwobidsonsingletaskswhosebidcostsaresmallest. Table3.6 showstheunnecessarybidssubmittedbyrobotsinthefirstroundofsequentialauctions withbundlesizetwoinTable3.5(tiesarebrokenarbitrarily). We now generalize this idea to sequential auctions with any given bundle size k and explain exactly which bundles robots need to bid on and why the resulting bids are 3 If there exist multiple bids with the smallest bid cost, then the two bids are any two of them. Other- wise, the two bids consist of the one with the smallest bid cost and any one of the bids with the second smallestbidcost. 4 We can prove this for Case 2 by contradiction: If the auctioneer allocates task x to robot r and the bid cost of robot r on task x is larger than those of the two bids with the smallest bid costs, then we can decrease the resulting robot cost of robot r as follows: assume that the auctioneer allocates the other winning task x ′ ̸= x to some other robot r ′ ̸= r, then we can find at least one task x ′′ with x ′′ ̸= x and x ′′ ̸= x ′ whose bid cost of robot r is smaller than its bid cost on task x. Thus, the robot cost r can be decreasediftheauctioneerallocatestaskx ′′ insteadoftaskxtorobotr. 78 Tasks BundleSize Bids Bids |U| k (calculated) (submitted) 20 20 1048575 1048575 (100.00%) 20 19 1048574 1047074 (99.86%) 20 18 1048554 1030781 (98.30%) 20 17 1048364 944747 (90.12%) 20 16 1047224 809850 (77.33%) 20 15 1042379 561204 (53.84%) 20 14 1026875 343120 (33.41%) 20 13 988115 184060 (18.63%) 20 12 910595 85684 (9.41%) 20 11 784625 34510 (4.40%) 20 10 616665 12130 (1.97%) 20 9 431909 3764 (0.87%) 20 8 263949 1048 (0.40%) 20 7 137979 315 (0.23%) 20 6 60459 105 (0.17%) 20 5 21699 39 (0.18%) 20 4 6195 16 (0.26%) 20 3 1350 7 (0.52%) 20 2 210 3 (1.43%) 20 1 20 1 (5.00%) Table3.7: ReductioninNumberofBidsSubmitted sufficient for an unchanged winner-determination phase to allocate the same additional tasks to the same robots during each round (modulo tie breaking). To determine which bids to submit in each round of a sequential auction with bundle sizek, we introduce a noveldatastructure, bid trees, that characterizesthelocal informationofrobots needed by the auctioneer in the winner-determination phase to allocate min(k,|U|) additional tasks to either the same robot or different robots so that the current team cost increases theleast. Each robot constructs k bid trees in the bid-construction phase, one for each 1≤ k ′ ≤ k. Each node of the k ′ th bid tree is labeled with a bundle of k ′ tasks. Each edge islabeledwithatask. Thek ′ thbidtreeisrecursivelyconstructed,startingwithitsroot. Consider any node in the bid tree. It is labeled with a bundle that has the smallest bid costamongallbundlesofk ′ tasksthatdonotcontainanyofthetasksthatlabeltheedges fromtheroottothenodeinquestion(tiescanbebrokenarbitrarily). Eachnodeatdepth min(k−k ′ ,|U|−k ′ )isaleaf. Thebidtreeisemptyifthisdepthisnegative. (Theterm |U|−k ′ covers the case where the number of tasks is smaller than the bundle size k.) 79 Otherwise, each edge from the node in question to itsk ′ children is labeled with one of the tasks contained in the bundle. This completes the construction of the bid trees. The robotthensubmitsbidsonallbundlesthatlabelnodesinitsbidtrees,calculatingthebid costsasbefore. (Thesamebundlecanlabelseveralnodesbut,ofcourse,resultsinonly one bid.) Thus, the number of bids per robot is bounded by a constant for sequential auctions with a given bundle size. Table 3.7 demonstrates the substantial reduction in the number of bids submitted by a robot for different bundle sizes in case there are 20 tasks. Theorem3.3.2 The winner-determination phase allocates the same min(k,|U|) addi- tionaltasks(modulotiebreaking)torobotsnomatterwhethertherobotssubmitbidson all non-empty bundles with at mostk tasks or submit bids on the bundles from their bid treesonly. Proof: Assume that the robots bid on all non-empty bundles with at most min(k,|U|) tasks and that the winning collection of bids is W. We show that we can achieve the same team cost if the robots bid on the bundles from their bid trees only. Obviously, the team cost cannot be smaller if the robots bid on the bundles from their bid trees only, a subset of all non-empty bundles with at most k tasks. We thus need to show that it cannot be larger either. To this end, consider an arbitrary winning bid b∈ W whose bundle b x is not in any bid tree of robot b r . We now show that we can replace this bidb with a bidb ′ so that the bundle of bidb ′ is in a bid tree, the collection of bids remains potentially winning, and the bid cost of bid b ′ is no larger than the bid costofbidb,whichimpliesthatthesubstitutioncannotincreasetheteamcostsincethe c team operator is monotonically increasing in its arguments. Similar substitutions can then be made for all bids in W, if necessary, until all bids are on bundles from the bid trees of the robots, which provesthe theorem. Toprovethe property, consider the tasks 80 1st bid tree for robot r 1 2nd bid tree for robot r 1 x 1 {x 1 , x 2 } x 2 x 1 depth 0 depth 1 BidTreesforRobotr 1 1st bid tree for robot r 2 2nd bid tree for robot r 2 x 2 {x 1 , x 2 } x 3 x 2 depth 0 depth 1 BidTreesforRobotr 2 Figure3.5: BidTreesforSequentialAuctionswithBundleSizeTwo F = ∪ b ′′ ∈W\{b} b ′′ x thatarepartofthewinningbidsexceptforbidb. Thefollowingprop- erties hold: b ′ is in a bid tree, b ′ r = b r (both bids are from the same robot),|b ′ x | =|b x | (both bids are on the same number of tasks), b ′ x ∩F =∅ (no task is allocated to more thanonerobot)andb ′ c ≤ b c (thebidcostofthenewbidisnolargerthanthebidcostof the previous bid). We achieve these properties by choosing the bidb ′ with the smallest bidcostamongallbidsfromthe|b x |thbidtreeofrobotb r whosebundlesdonotcontain tasks in F, as follows: We start at the root of the|b x |th bid tree of robot b r as current node. If the bundle of the current node does not contain any tasks in F, then we pick thebidonthisbundle. Otherwise,wechooseataskthatiscontainedinboththisbundle andF (tiescanbebrokenarbitrarily)andfollowtheedgelabeledwiththistaskfromthe current node to one of its children, and repeat the procedure. This way, we are guaran- teed to reach a node whose bundle does not contain any tasks inF becauseF contains min(k−|b x |,|U|−|b x |)tasksandthe|b x |thbidtreehasdepthmin(k−|b x |,|U|−|b x |). The bundle of this node is the bundle with the smallest bid cost that does not contain anytasksinF andhasthestatedproperties,perconstructionofthebidtrees. Consideragainthemulti-robotroutingproblemshowninFigure3.1fortheMiniMax teamobjective. Figure3.5showsthebidtreesofrobotsr 1 andr 2 duringthefirstround of a sequential auction with bundles size two. The bids on the bundles contained in thesebidtreesaretheexactlysameasthoseinTable3.6thatarenotcrossedout. Figure 3.6showsthebidtreesofrobotsr 1 andr 2 duringthefirstroundofasequentialauction 81 1st bid tree for robot r 1 2nd bid tree for robot r 1 3rd bid tree for robot r 1 x 1 {x 1 , x 2 } {x 1 , x 2 , x 3 } x 2 x 3 x 1 x 2 x 1 x 2 {x 2 , x 3 } {x 1 , x 3 } depth 0 depth 1 depth 2 BidTreesforRobotr 1 1st bid tree for robot r 2 2nd bid tree for robot r 2 3rd bid tree for robot r 2 x 2 {x 1 , x 2 } {x 1 , x 2 , x 3 } x 3 x 1 x 2 x 3 x 1 x 2 {x 2 , x 3 } {x 1 , x 3 } depth 0 depth 1 depth 2 BidTreesforRobotr 2 Figure3.6: BidTreesforSequentialAuctionswithBundleSizeThree with bundle size three. For example, robotr 1 constructs its second bid tree as follows: {x 1 ,x 2 }isthebundleoftwotargetswiththesmallestbidcostandthusistherootofthe bid tree,{x 2 ,x 3 } is the bundle of two targets different from targetx 1 with the smallest bid cost and thus is a child of the root of the bid tree, and finally,{x 1 ,x 3 } is the bundle oftwotargetsdifferentfromtargetx 2 withthesmallestbidcostandthusisanotherchild oftherootofthebidtree. 3.3.3 Winner-DeterminationPhase Althoughwemanagedtoreducethenumberofbids,itisnotclearatthispointwhether the auctioneer can determine the winning bids efficiently. In the following, we show that the auctioneer can determine the winning bids with a runtime that is linear in the number of bids and thus linear in the number of robots and independent of the number oftasksforsequentialauctionswithagivenbundlesize. Theorem3.3.3 The auctioneer in sequential auctions with a given bundle size can determinethewinningbidswitharuntimethatislinearinthenumberofbids. Proof: LetB bethebidsoftherobotsonthebundlesfromtheirbidtrees. Theauc- tioneerneedstoallocatemin(k,|U|)additionaltaskstorobots. ItfirstfindsallO(1)dif- ferentnon-decreasingsequencesofatmost|R|positiveintegersthatsumtomin(k,|U|). 82 For example, there are two such sequences for min(k,|U|) = 3 and|R| = 2, namely the sequence (1,2) and the sequence (3). A portfolio for such a sequence (s(1)...s(l)) is a collection of bids (b(1)...b(l)) withb(i)∈ B and|b(i) x | = s(i) for all 1≤ i≤ l. Theportfolioisconsistentiffb(i) r ̸= b(j) r andb(i) x ∩b(j) x =∅forall 1≤ i < j≤ l. Note that a consistent portfolio is a potentially winning collection of bids. To find the winning collection of bids, the auctioneer constructs a search tree for each sequence. Eachnodeofthesearchtreeforagivensequence(s(1)...s(l))islabeledwithaportfo- lioforthesequence. Eachedgeislabeledwithaconstraint,whichcanbeoftwokinds: 1) “x∈ b(j) x ” for a given x∈ U and a given 1≤ j ≤ l and 2) “b(j) r = b r and b(j) x = b x ” for a given b∈ B with|b x | = s(j) and a given 1≤ j≤ l. The search treeforagivensequence (s(1)...s(l))isrecursivelyconstructed,startingwithitsroot. Consider any node in the search tree. It is labeled with a portfolio (b(1)...b(l)) with thesmallestevaluationcostamongallportfoliosthatsatisfytheconstraintsthatlabelthe edges from the root to the node in question, as follows: If the constraint is “x∈ b(j) x ” then the portfolio needs to satisfyx∈ b(i) x fori = j and, for all 1≤ i≤ l withi̸= j, x̸∈ b(i) x . If the constraint is “b(j) r = b r and b(j) x = b x ,” then the portfolio needs to satisfyb(i) r = b r andb(i) x = b x fori = j and, for all 1≤ i≤ l withi̸= j,b(i) r ̸= b r andb(i) x ∩b x =∅. Thenodeisdeletedfromthesearchtreeifnoportfoliosatisfiesthe constraints. Each node with a consistent portfolio is a leaf. Otherwise, the auctioneer chooses any 1≤ i≤ l with b(i) r = b(j) r or b(i) x ∩b(j) x ̸=∅ for some 1≤ j≤ l withi̸= j to generate the constraints that label the edges from the node in question to its children: 1) “x∈ b(j) x ,” one constraint for each 1≤ j≤ l with j̸= i and each x∈ b(i) x ; and 2) “b(j) r = b r and b(j) x = b x ,” one constraint for each 1≤ j≤ l and eachb∈B withb(i) r =b r and|b x | =s(j). Thejustificationfortheseconstraintsisthat atleastoneofthemmustbesatisfied: Eithersomebidbyrobotb(i) r ispartoftheport- folio or there is a bid part of the portfolio whose bundle includes some task x∈ b(i) x . 83 (r 2 ,{x 2 }) (r 1 , {x 1 }) Search Tree for (1, 1) (r 2 , {x 1 , x 2 }) Search Tree for (2) 4 4 Figure3.7: SearchTreesforSequentialAuctionswithBidTreesandBundleSizeTwo (Ifthiswasnotthecase,thenonecouldsubstitutebidb(i)forthebidattheithposition, resultinginapotentiallywinningbidwithoutincreasingtheevaluationcost.) Thiscom- pletes the construction of the search trees. The winning collection of bids then is any portfolio(b(1)...b(l))withthesmallestevaluationcostamongallconsistentportfolios thatlabelnodesinthesearchtrees,perconstructionofthesearchtrees. ThereareO(1) sequencesandthusO(1)searchtrees. Thedepthofthesearchtreeforagivensequence (s(1)...s(l))isatmostk+landthusO(1)sinceeachconstraintimposesanadditional restriction on a portfolio and no portfolio can satisfy more than k +l constraints. The numberofchildrenofeverynodeineachsearchtreeisO(1). Thus,eachsearchtreehas O(1) O(1) = O(1) nodes. The portfolio of each node can be determined in timeO(|B|) by finding the bid with the smallest bid cost that satisfies the constraints independently foreachpositionintheportfolio. Consequently,theruntimeofwinnerdeterminationis O(1)×O(1)×O(|B|) =O(|B|),asclaimed. Our winner-determination rule can be further optimized for given bundle sizes. For example, the auctioneer constructs only 13 potentially winning collections of bids and chooses the best one among them in the winner-determination algorithm developed in [DK09]forsequentialauctionswithbidtreesandbundlesizethree. Consider again the multi-robot routing problem shown in Figure 3.1 for the Mini- Maxteamobjective. Figure3.7showsthecorrespondingsearchtreesforthefirstround of a sequential auctions with bid trees and bundle size two. Figure 3.8 shows the cor- responding search trees for the first round of a sequential auction with bid trees and 84 (r 2 ,{x 2 }) (r 2 , {x 1 , x 2 }) (r 2 ,{x 1 }) (r 1 , {x 2 , x 3 }) (r 2 ,{x 2 }) (r 1 , {x 1 , x 3 }) (r 2 ,{x 3 }) (r 1 , {x 1 , x 2 }) (r 2 ,{x 3 }) (r 2 , {x 1 , x 2 }) (r 1 ,{x 3 }) (r 2 , {x 1 , x 2 }) (r 1 ,{x 2 }) (r 2 , {x 1 , x 3 }) (r 1 ,{x 1 }) (r 2 , {x 2 , x 3 }) (r 2 ,{x 1 }) (r 1 , {x 2 , x 3 }) (r 2 ,{x 3 }) (r 1 , {x 1 , x 2 }) (r 2 ,{x 1 }) (r 2 , {x 2 , x 3 }) (r 1 ,{x 3 }) (r 2 , {x 1 , x 2 }) (r 1 ,{x 1 }) (r 2 , {x 2 , x 3 }) (r 2 ,{x 1 }) (r 1 , {x 2 , x 3 }) (r 1 ,{x 1 }) (r 2 , {x 2 , x 3 }) b(1) r = r 2 and b(1) x = {x 2 } b(1) r = r 2 and b(1) x = {x 3 } b(1) r = r 2 and b(1) x = {x 1 } x 2∈ b(2) x b(2) r = r 2 and b(2) x = {x 1 ,x 2} b(2) r = r 2 and b(2) x = {x 1 ,x 3 } b(2) r = r 2 and b(2) x = {x 2 ,x 3} b(2) r = r 2 and b(2) x = {x 1 ,x 2 } b(1) r = r 2 and b(1) x = {x 1 } b(1) r = r 2 and b(1) x = {x 3 } x 3∈ b(2) x b(2) r = r 2 and b(2) x = {x 2 ,x 3 } b(1) r = r 2 and b(1) x = {x 1 } b(2) r = r 2 and b(2) x = {x 2, x 3 } Search Tree for (1, 2) (r 1 , {x 1 , x 2 , x 3 }) Search Tree for (3) 12 12 6 12 12 8 12 6 12 8 12 8 12 Figure3.8: SearchTreesforSequentialAuctionswithBidTreesandBundleSizeThree bundle size three. The underlined bids were used to generate the constraints, and con- sistent portfolios are annotated with their evaluation costs. Table 3.8 tabulates the bids calculated and submitted by robots and the allocations the auctioneer needs to evaluate insequentialauctionswithbidtreesandbundlesizetwo. In the following sections, robots in sequential auctions with bundles always use bid treestoconstructtheirbids,soweleaveoutthenotion”bidtrees”insequentialauctions withbundlesandbidtressforreadability. 3.3.4 Refinement The auctioneer can be cautious and allocate only one additional task of the min(k,|U|) tasks per round, namely one with the smallest bid cost on single tasks. In this case, the bid-construction phase changes as follows: Every robot also needs to bid on all tasks thatarepartoflargerbundlesthatitbidsonsincetheauctioneerneedstheirbidcostson single tasks to select the winning task. This increases its number of bids by a constant thatisusuallysmall. Asexample,consideragainthemulti-robotroutingproblemshown 85 BidConstruction WinnerDetermination Bids(calculated) Bids(submitted) Allocations(evaluated) Decision Round1 Robotr 1 : 4onbundle{x 1 } Robotr 1 : 4onbundle{x 1 } 6onbundle{x 2 } 6onbundle{x 2 } {x 1 ,x 2 }→r 2 : 4 12onbundle{x 3 } 6onbundle{x 1 ,x 2 } 6onbundle{x 1 ,x 2 } Theauctioneer 12onbundle{x 1 ,x 3 } allocatestargets 12onbundle{x 2 ,x 3 } x 1 andx 2 to robotr 2 (by Robotr 2 : 4onbundle{x 1 } Robotr 2 : 2onbundle{x 2 } (x 1 →r 1 ,x 2 →r 2 ): 4 tiebreaking) 2onbundle{x 2 } 4onbundle{x 3 } 4onbundle{x 3 } 4onbundle{x 1 ,x 2 } 4onbundle{x 1 ,x 2 } 12onbundle{x 1 ,x 3 } 8onbundle{x 2 ,x 3 } Round2 Robotr 1 : 12onbundle{x 3 } Robotr 1 : 12onbundle{x 3 } (x 3 →r 1 ): 12 Theauctioneer allocatestarget Robotr 2 : 12onbundle{x 3 } Robotr 2 : 12onbundle{x 3 } (x 3 →r 2 ): 12 x 3 torobotr 2 (bytiebreaking) Table 3.8: Sequential Auctions with Bid Trees and Bundle Size Two for Example in Figure3.1(MiniMaxTeamObjective) in Figure 3.1 for the MiniMax team objective. Figures 3.5 and 3.6 show that robot r 1 already bids on all targets that are part of larger bundles that it bids on during the first round of sequential auctions with bundle sizes two and three and thus does not need to submit any additional bids during the first round. However, robotr 2 has to submit one additional bid on target x 1 during the first round of sequential auctions with bundle size two. The winner-determination phase changes as follows: The auctioneer first determines the winning collection W of bids, as before. Then, it determines the bid b∈ W and taskx∈ b x withc robot br ({x})≤ c robot b ′ r ({x ′ }) for allb ′ ∈ W andx ′ ∈ b ′ x (the taskwiththesmallestbidcostamongallwinningtasksinW). Itdoesthisbycomparing the bid of robotb ′ r on taskx ′ for allb ′ ∈ W andx ′ ∈ b ′ x and choosing the one with the smallest bid cost (ties can be broken arbitrarily). It then allocates only this additional taskxtothebiddingrobotb r . Consideragainthemulti-robotroutingproblemshowninFigure3.1fortheMiniMax team objective. Table 3.9 tabulates the bids calculated and submitted by robots and the 86 BidConstruction WinnerDetermination Bids(calculated) Bids(submitted) Allocations(evaluated) Decision Round1 Robotr 1 : 4onbundle{x 1 } Robotr 1 : 4onbundle{x 1 } 6onbundle{x 2 } 6onbundle{x 2 } ({x 1 ,x 2 }→r 2 ): 4 12onbundle{x 3 } 6onbundle{x 1 ,x 2 } (x 1 →r 2 ): 4 6onbundle{x 1 ,x 2 } (x 2 →r 2 ): 2 Theauctioneer 12onbundle{x 1 ,x 3 } allocatestarget 12onbundle{x 2 ,x 3 } x 2 torobotr 2 (bytiebreaking) Robotr 2 : 4onbundle{x 1 } Robotr 2 : 4onbundle{x 1 } (x 1 →r 1 ,x 2 →r 2 ): 4 2onbundle{x 2 } 2onbundle{x 2 } 4onbundle{x 3 } 4onbundle{x 3 } 4onbundle{x 1 ,x 2 } 4onbundle{x 1 ,x 2 } 12onbundle{x 1 ,x 3 } 8onbundle{x 2 ,x 3 } Round2 Robotr 1 : 4onbundle{x 1 } Robotr 1 : 4onbundle{x 1 } ({x 1 ,x 3 }→r 2 ): 12 12onbundle{x 3 } 12onbundle{x 3 } 12onbundle{x 1 ,x 3 } 12onbundle{x 1 ,x 3 } (x 1 →r 1 ,x 3 →r 2 ): 8 Theauctioneer (x 1 →r 2 ): 4 allocatestarget Robotr 2 : 4onbundle{x 1 } Robotr 2 : 4onbundle{x 1 } (x 3 →r 2 ): 8 x 1 torobotr 1 8onbundle{x 3 } 8onbundle{x 3 } 12onbundle{x 1 ,x 3 } 12onbundle{x 1 ,x 3 } Round3 Robotr 1 : 12onbundle{x 3 } Robotr 1 : 12onbundle{x 3 } Theauctioneer (x 3 →r 1 ): 12 allocatestarget Robotr 2 : 8onbundle{x 3 } Robotr 2 : 8onbundle{x 3 } (x 3 →r 2 ): 8 x 3 torobotr 2 Table 3.9: Sequential Auctions with Bundle Size Two and Refinement for Example in Figure3.1(MiniMaxTeamObjective) allocationstheauctioneerneedstoevaluateinasequentialauctionwithbundlesizetwo andrefinement. Figure3.9showstheresultingallocations. 3.4 ExperimentalEvaluation We now evaluate the performance of sequential auctions with bundles empirically for solvingsimple-taskallocationproblemsinthedomainofmulti-robotrouting. 3.4.1 ExperimentalSettings Wecomparethefollowingsimple-taskallocationalgorithms: • AuctionAlgorithms: 87 r 1 x 1 x 2 r 2 x 3 TeamCost=8 Figure 3.9: Allocations of Sequential Auctions with Bundle Size Two and Refinement forExampleinFigure3.1(MiniMaxTeamObjective) – Parallel: This algorithm implements parallel auctions described in Subsec- tion3.2.2. – SAB (k = 1): This algorithm implements sequential auctions with bundle size one, which is identical to sequential single-item auctions described in Subsection3.2.3. – SAB (k = 2): This algorithm implements sequential auctions with bundle sizetwo. – SAB (k = 3): This algorithm implements sequential auctions with bundle sizethree. Unless explicitly mentioned otherwise, all SAB algorithms with k > 1 are the oneswithrefinementinourexperiments. • CentralizedAlgorithms: – Optimal: This algorithm implements an optimal algorithm. In order to cal- culate the smallest team costs of combinatorial auctions described in Sub- section 3.2.1 and serve as a gold standard for comparing team costs of 88 other simple-task allocation algorithms, we formulate the multi-robot rout- ingproblemasamixedintegerprogram(MIP)andsolveitviathecommer- cialsoftwareCPLEX[Cpl01]. – Rollout: This algorithm implements a centralized rollout algorithm intro- duced in [ZKT06]. In a rollout algorithm, targets are allocated to robots in multiple rounds. During each round, the central planner allocates one additional target to some robot as follows: It first constructs all possible single-targetallocationsthateachallocateoneadditionaltargettooneeligi- ble robot. It then evaluates each single-target allocation by first tentatively performingthesingle-targetallocationandthenusingsequentialsingle-item auctionstogreedilycompletetheresultingpartialsolution. Finally,thecen- tral planner chooses the single-target allocation as the winner whose team costofthegreedilycompletedsolutionissmallest. – Random: This algorithm implements a randomized task-allocation algo- rithm. Inarandomizedalgorithm,eachtargetisrandomlyallocatedtosome robotaslongastherobotiseligibleforvisitingit. • Distributed: This algorithm implements a simple distributed algorithm intro- duced in [Dia04]. In a distributed algorithm, each robot independently schedules its own route to visit all targets that have not been visited. Each robot informs the other robots whenever it has visited a target so that other robots can remove thattargetfromtheircurrentroutesandre-scheduletheirroutesaccordingly. This algorithmisalsousedin[Sar07]formulti-robotexploration. We evaluate our task-allocation algorithms for solving multi-robot routing prob- lems on four-neighbor grids of size 51× 51 with square cells that either blocked or unblocked. Figure 3.10 shows three different kinds of terrain used in the experiments. 89 Outdoor-LikeTerrain EmptyTerrain Indoor-LikeTerrain Figure 3.10: Experimental Test-Beds for Multi-Robot Routing: Outdoor-Like Terrain, EmptyTerrain,andIndoor-LikeTerrain Thefirstkindisanoutdoor-liketerrainwhereblockedcellsarerandomlyremoveduntil thedensity of blockedcells dropsto 10 percent, resulting in terrain with randomobsta- cles. The second kind is an empty terrain with no blocked cells. The third kind is an indoor-like terrain which resembles an office environment with walls and doors. The positionofthewallsanddoorsarefixed,butdoorsareclosedwith30percentprobabil- ity. Weevaluateourtask-allocationalgorithmsforsolvingmulti-robotroutingproblems inbothknownandunknownterrain: • Known terrain: When the terrain is known, robots initially know the blockage status of each cell of the given terrain. (The blockage status of a cell is either blocked or unblocked.) After targets have been allocated to robots, each robot visits its allocated targets without the need of updating the blockage status of cells in the terrain. Thus, the task-allocation algorithm is executed only once. In 90 this case, we report the resulting team cost and the response time 5 of the task- allocationalgorithm. Unlessexplicitlymentionedotherwise,theterrainisknown totherobotsinourexperiments. • Unknown terrain: When the terrain is unknown, robots initially do not know the blockage status of the cells (except for the neighbor cells of their initially locations). Robots first assume that all cells are unblocked. 6 When robots move from one cell to another, they sense and update the blockage status of the cells that are neighbors of their currently located cells. Whenever a robot discovers new knowledge of the terrain, all robots re-calculate their robot costs for visiting their currently allocated targets. If any robot cost changes, the task-allocation algorithm is executed again to allocate all un-visited targets to robots. Thus, the task-allocationalgorithmmightbeexecutedrepeatedlyinunknownterrain. Inthis case,wereporttheresultingteamcost,thenumberoftherepeatedexecutions,and thecumulativeresponsetime 7 ofthetask-allocationalgorithm. Weevaluateourtask-allocationalgorithmsforsolvingmulti-robotroutingproblems with different numbers of robots and targets. The size of a multi-robot routing problem is its total number of robots and targets. In our experiments, the initial locations of the robots and the locations of the targets are randomly chosen from the unblocked cells of the terrain, and the locations of the targets are guaranteed to be reachable from the initiallocationsoftherobots(sothateveryrobotcanreacheachtarget). Inbothknown 5 Although robots in an auction algorithm can calculate their bids in parallel, we report the total com- putation times of all robots and the auctioneer as the response time of the auction algorithm in order to have consistent comparisons with the other algorithms (especially the centralized ones), which is a commonpracticeintheliterature[DS03]. 6 Thisisknownasthefreespaceassumption,whichhasbeenwidelyusedinincrementalsearch[KS96, KL02]. 7 Thecumulativeresponsetimeisthesumofresponsetimesfromtherepeatedexecutionsofthetask- allocationalgorithm. 91 and unknown terrain, robots always know the locations of targets. We assume that multiplerobotsareabletooccupythesamesmallcellandneverblockeachotherinour experiments 8 . Giventhattherobotsandtargetsarecharacterizedbyexactlyonecapability,amulti- robot routing problem with different capabilities can always be decomposed into mul- tiple multi-robot routing problems with the same capability that are then solved inde- pendently. In our experiments, we evaluate our task-allocation algorithms for solving multi-robotroutingproblemswithonlyonecapability. Amulti-robotroutingproblemwithonlyonecapabilityisguaranteedtohaveasolu- tion as long as the sum of the capacities of the robots is not smaller than the number of targets. Inourexperiments,weevaluateourtask-allocationalgorithmswiththefollow- ingdifferentrobot-capacityconstraints: • Tight Robot-Capacity Constraint: The sum of the capacities of robots is equal to the number of targets. Unless explicitly mentioned otherwise, the robot- capacityconstraintistightinourexperiments. • LooseRobot-CapacityConstraint: Thesumofthecapacitiesofrobotsisgreater thanthenumberoftargets. In order to determine its robot cost in a multi-robot routing problem, a robot needs tosolveaversionofatravelingsalespersonproblem[GP02]forboththeMiniSumand MiniMax team objectives when calculating the smallest travel distance needed to visit its allocated targets, and a version of a traveling repairman problem [ACP + 86] for the MiniLat team objectives when calculating the smallest cumulative latencies needed to visititsallocatedtargets. Inourexperiments,weevaluateourtask-allocationalgorithms withthefollowingdifferentapproachesofrobot-costcalculations: 8 Thisisacommonassumptionmadeinexperimentsofmulti-robotrouting[ZKT06,Sar07]andmulti- robotcoverage[ZJKK05]. 92 Results Terrain Blockage Capacities Robots Targets Optimal Robot-Cost Robot-Capacity Instances Status Algorithm Calculation Constraint TableA.1 TeamCosts Outdoor-Like Known 3 2-10 6-30 Yes Approximated Tight 15 TableA.2 TeamCosts Outdoor-Like Known 3 2-10 6-30 No Approximated Tight 100 TableA.3 ResponseTimes Outdoor-Like Known 3 2-10 6-30 No Approximated Tight 100 TableA.4 Combined(CostsandTimes) Outdoor-Like Known 3 2-10 6-30 No Approximated Tight 100 TableA.5 TeamCosts Empty Known 3 2-10 6-30 No Approximated Tight 100 TableA.6 ResponseTimes Empty Known 3 2-10 6-30 No Approximated Tight 100 TableA.7 TeamCosts Office-Like Known 3 2-10 6-30 No Approximated Tight 100 TableA.8 ResponseTimes Office-Like Known 3 2-10 6-30 No Approximated Tight 100 TableA.9 TeamCosts Outdoor-Like Known 2-5 10 20-50 No Approximated Tight 100 TableA.10 ResponseTimes Outdoor-Like Known 2-5 10 20-50 No Approximated Tight 100 TableA.11 TeamCosts Outdoor-Like Known 2-5 10 20-50 No Ideal Tight 100 TableA.12 ResponseTimes Outdoor-Like Known 2-5 10 20-50 No Ideal Tight 100 TableA.13 TeamCosts Outdoor-Like Known 3-∞ 10 30 No Approximated Loose 100 TableA.14 ResponseTimes Outdoor-Like Known 3-∞ 10 30 No Approximated Loose 100 TableA.15 TeamCosts Outdoor-Like Unknown 3 2-10 6-30 No Approximated Tight 100 TableA.16 RepeatedExecutions Outdoor-Like Unknown 3 2-10 6-30 No Approximated Tight 100 TableA.17 ResponseTimes Outdoor-Like Unknown 3 2-10 6-30 No Approximated Tight 100 Table3.10: SummaryofSimple-TaskAuctionExperiments • Approximated Robot-Cost Calculation: Robots use the two-opt heuristic [LLKS85]tocalculatetheapproximatedrobotcostsofvisitingtheirallocatedtar- gets. Unlessexplicitlymentionedotherwise,therobot-costcalculationisapprox- imatedinourexperiments. • IdealRobot-CostCalculation: Robotsevaluatesallpossibleroutestovisittheir allocatedtargetsfromitscurrentlocationtofindthesmallestrobotcosts. 3.4.2 ExperimentalResults Weperformexperimentswithdifferentsettingsintroducedintheprevioussubsectionto evaluateoursimple-taskauctionalgorithms,assummarizedinTable3.10. Thedetailed experimental data and explanations can be found in Appendex A.1. We make the fol- lowingobservations: TeamCosts • The team costs of sequential auctions with larger bundle size are smaller than thoseofsequentialauctionswithsmallerbundlesize. Thisobservationmeetsour expectation since sequential auctions with the larger bundle size consider more synergiesamongtargetsandthusreducetheteamcosts. 93 • The team costs of sequential auctions with bundles are smaller than those of par- allelauctions,therandomizedalgorithm,andthedistributedalgorithm. • The team costs of sequential auctions with bundles are larger than the rollout algorithm. This observation is not surprising since sequential auctions with a smallbundlesizeevaluatepartialsolutionstomakeallocationdecisions,whichis muchmoregreedythantherolloutalgorithmthatevaluatescompletesolutionsto makeallocationdecisions. • The team costs of sequential auctions with bundle size three are still about 10-20 percent away from the minimal team costs for the MiniSum and MiniLat team objectives. The team costs may be up to 60 percent away from the minimal ones fortheMiniMaxteamobjective. Thisobservationinspiresustodevelopnegotia- tion algorithms in the next chapter to improve the initial allocations produced by sequentialauctionswithbundles. ResponseTimes • The response times of the optimal algorithm quickly increase with the problem size. Forexample,wearenotabletodeterminetheminimalteamcostsforanyof the15instanceswith10robotsand40targetswithintheone-hourruntimelimit. • The response times of the rollout algorithm are larger than those of the other task-allocationalgorithms(exceptfortheoptimalalgorithm)andincreasequickly with the problem size. This observation meets our expectation since the rollout algorithm has to complete a partial solution in order to evaluate a single target- allocation. • Theresponsetimesoftherandomizedalgorithm,parallelauctions,andsequential auctionswithsmallbundlesizes(k=1,2)aresmall(lessthan5seconds). 94 • Theresponsetimesofsequentialauctionswithbundlesincreasesquicklywiththe bundlesize. Thisobservationmeetsourexpectationsincethenumberofthetask- allocations robots and the auctioneer need to evaluate grows exponentially with thebundlesizeinsequentialauctionswithbundles. 3.5 Summary In this chapter, we studied auction algorithms for solving simple-task allocation prob- lems. We developed a new auction algorithm, called sequential auctions with bundles, that allocates a given number of unallocated tasks to robots in each round so that the team cost increases the least. Our theoretical results show that the number of bids per robot per round is only a constant number and the runtime of the winner-determination fortheauctioneerisonlylinearinthenumberofbiddersandindependentofthenumber oftotalunallocatedtasks. Ourexperimentalresultsshowthatsequentialauctionswitha smallbundlesize(greaterthanone)canreducetheteamcostsofsequentialsingle-item auctionswhilemaintainingtheadvantageofsmallresponsetimes. 95 Chapter4 Simple-TaskNegotiations Inthischapter,westudytask-allocationalgorithmsthatimplementthemulti-partynego- tiationprotocolshowninFigure1.10forsolvingsimple-taskallocationproblems. Inall auctionalgorithmsstudiedinthepreviouschapter,robotsbidonlyonunallocatedtasks, and tasks cannot be re-allocated once they have been allocated to robots. In this chap- ter,adifferentperspectiveistakenforsolvingsimple-taskallocationproblems,namely, robots negotiate with each other to exchange their allocated tasks in order to improve their current allocations. Such negotiations are usually studied in a competitive set- ting where robots are self-interested and a contract is accepted only if all participants are better off from the contract [GMR97a, TRS04, San98, AS99]. In this dissertation, robots are cooperative and always collaborate to minimize the team cost. Thus, they shouldconsideralltaskexchangesthatdecreasethecurrentteamcost. Weusetheterm “negotiation” here to describe the interaction of cooperative robots as specified in the multi-partynegotiationprotocol. Many existing negotiation algorithms [GMR97a, DS00] repeatedly perform single task exchanges 1 that each decrease the team cost of the current solution. However, robotstakeintoaccountonlythesynergiesbetweensingletasksthatarenotallocatedto them and their allocated tasks when they evaluate single task exchanges. Thus, negoti- ationalgorithmsbasedonsingletaskexchangesusuallygeteasilystuckinlocaloptima andcannotreducetheteamcostsubstantially[GMR97a]. Inordertominimizetheteam 1 Ataskexchangeisabasicnegotiationcontractbetweentworobotsonsingletasks. Iteithertransfers onlyasingletaskfromonerobottoanother(or,synonymously,O-contract[San98]),orswapstwosingle tasksbetweenrobots(or,synonymously,S-contract[San98]). 96 cost, however, it might be necessary to perform multiple task exchanges that decrease theteamcost[DS02,San98]ofthecurrentsolution,wheremultipletaskexchangescan transfermultipletasksamongmultiplerobotsatthesametime. In this chapter, we illustrate that negotiation algorithms based on single task exchanges can be extended to consider more synergies among tasks and result in better allocations by performing multiple task exchanges among robots in each round. Our goal is to develop negotiation algorithms for solving simple-task allocation problems (simple-tasknegotiations)thatachievethefollowingmyopicoptimizationperformance principleonmultipletasks: Performance principle of simple-task negotiations: To perform upto a givennumberoftaskexchangesamongrobotsineachroundofsimple-task negotiationssothatthecurrentteamcostdecreasesthemostpossible. We proceed as follows: Section 4.1 formalizes and exemplifies the problem studied in this chapter. Section 4.2 summarizes several existing simple-task negotiation algo- rithms. In Section 4.3, we first introduce a novel data structure, called partialk-swaps, thatsuccinctlycharacterizesthelocalinformationofrobotsnecessarytoachievetheper- formanceprincipleofsimple-tasknegotiations. Afterthat,wedevelopanewnegotiation algorithm, called sequential negotiations with K-swaps, in which robots construct and propose partial k-swaps to each other and perform a given number of task exchanges among them in each round so that the current team cost decreases the most. Finally, Section4.4presentsexperimentalresults,andSection4.5concludesthischapter. 4.1 ProblemDescription Wecontinuetostudysimple-taskallocationproblemsasformalizedinSection3.1. Let (X r ) r∈R = (X r 1 ,...,X rm ) be a complete solution of a given simple-task allocation 97 r 2 x 2 x 4 r 3 x 3 r 1 x 1 Figure4.1: Example6ofMulti-RobotRouting {x 1 } {x 2 } {x 3 } {x 4 } {x 1 ,x 2 } {x 1 ,x 3 } {x 1 ,x 4 } {x 2 ,x 3 } r 1 1 7 7 1 7 9 3 21 r 2 5 3 11 5 9 13 7 17 r 3 5 11 3 5 11 11 7 17 {x 2 ,x 4 } {x 3 ,x 4 } {x 1 ,x 2 ,x 3 } {x 1 ,x 2 ,x 4 } {x 1 ,x 3 ,x 4 } {x 2 ,x 3 ,x 4 } {x 1 ,x 2 ,x 3 ,x 4 } r 1 9 7 21 9 9 21 21 r 2 11 11 17 11 13 17 17 r 3 13 9 17 13 11 17 17 Table4.1: RobotCostsforExampleinFigure4.1(MiniMaxTeamObjective) problem⟨R,X,E⟩. In this chapter, we study how robots can exchange their allocated tasks in (X r ) r∈R to produce a new complete solution ( X r ) r∈R = ( X r 1 ,..., X rm ) so that the team cost of the new allocations ( X r ) r∈R is smaller than that of the initial allocations (X r ) r∈R . Figure4.1showsamulti-robotroutingproblemwithsimpletargetsinknownterrain for the MiniMax team objective, which we use throughout this chapter to illustrate our negotiation algorithms. There exists only one capability. Robots r 1 , r 2 and r 3 provide this capability, and targets x 1 , x 2 , x 3 and x 4 require it. The capacities of all robots are 4,andthustheyareeligibleforallpossiblebundlesoftargetsinthisexample. Table4.1 tabulates the robot cost (the smallest travel time) of each robot for visiting any bundle oftargets. TheleftpartofFigure4.2showstheoptimalallocationsofthisexamplewith thesmallestteamcost3,whererobotr 2 isallocatedtargetx 2 ,robotr 3 isallocatedtarget x 3 , and robot r 1 is allocated targets x 1 and x 4 . Let (X r 1 =∅,X r 2 ={x 1 ,x 2 },X r 3 = 98 r 2 x 2 x 4 r 3 x 3 r 1 x 1 OptimalAllocations (TeamCost=3) r 2 x 2 x 4 r 3 x 3 r 1 x 1 InitialAllocations (TeamCost=9) Figure4.2: AllocationsforExampleinFigure4.1(MiniMaxTeamObjective) {x 3 ,x 4 }) be the initial allocations of this example with team cost 9, as shown in the rightpartofFigure4.2. 4.2 ExistingApproachestoSimple-TaskNegotiations In this section, we discuss several existing negotiation algorithms in the literature that performtaskexchangesamongrobots. The majority of existing negotiation algorithms for solving simple-task allocation problems are using the idea of either single-item transfers [LBK + 04, San98] or single- taskswaps[GMR97a,DS00],whereasingle-itemtransfermovesexactlyonetaskfrom one robot to another robot, and a single-task swap moves one task from one robot r to another robot r ′ and a different task from robot r ′ to robot r at the same time. In this dissertation, both single-item transfers and single-task swaps can be represented as singletaskexchangesbetweentworobots. Single task exchanges are simple to implement since each of them involves only two robots and at most two tasks. However, negotiation algorithms based on single task exchanges are not always able to decrease the team cost of the initial allocations. Consider again the multi-robot routing problem shown in Figure 4.1 for the MiniMax teamobjective. Table4.2tabulatesallpossiblesingletaskexchangesbetweenrobotsfor 99 Initiator Participant TaskfromInitiator TaskfromParticipant NewAllocations TeamCost toParticipant toInitiator r 1 r 2 ∅ {x 1 } ({x 1 },{x 2 },{x 3 ,x 4 }) 9 r 1 r 2 ∅ {x 2 } ({x 2 },{x 1 },{x 3 ,x 4 }) 9 r 1 r 3 ∅ {x 3 } ({x 3 },{x 1 ,x 2 },{x 4 }) 9 r 1 r 3 ∅ {x 4 } ({x 4 },{x 1 ,x 2 },{x 3 }) 9 r 2 r 3 {x 1 } ∅ (∅,{x 2 },{x 1 ,x 3 ,x 4 }) 11 r 2 r 3 {x 1 } {x 3 } (∅,{x 2 ,x 3 },{x 1 ,x 4 }) 17 r 2 r 3 {x 1 } {x 4 } (∅,{x 2 ,x 4 },{x 1 ,x 3 }) 11 r 2 r 3 {x 2 } ∅ (∅,{x 1 },{x 2 ,x 3 ,x 4 }) 17 r 2 r 3 {x 2 } {x 3 } (∅,{x 1 ,x 3 },{x 2 ,x 4 }) 13 r 2 r 3 {x 2 } {x 4 } (∅,{x 1 ,x 4 },{x 2 ,x 3 }) 17 r 2 r 3 ∅ {x 3 } (∅,{x 1 ,x 2 ,x 3 },{x 4 }) 17 r 2 r 3 ∅ {x 4 } (∅,{x 1 ,x 2 ,x 4 },{x 3 }) 11 Table4.2: SingleTaskExchangesforExampleinFigure4.1(MiniMaxTeamObjective) the given initial allocations (X r 1 =∅,X r 2 ={x 1 ,x 2 },X r 3 ={x 3 ,x 4 }) with team cost 9. 2 Noneofthesetaskexchangesareusefulsincetheteamcostoftheinitialallocations doesnotdecreaseafterperforminganysingleoneofthesetaskexchanges. Researchers have recognized that negotiation contracts involving multiple task exchangesareimportanttodecreasetheteamcostoftheinitialallocations, wheremul- tiple task exchanges move multiple tasks among multiple robots at the same time. For example,DiasandStentz[DS02,Dia04]proposetousecombinatorialexchanges,where robots in a combinatorial exchange can perform any number of task exchanges among them. Consideragainthemulti-robotroutingproblemshowninFigure4.1fortheMin- iMaxteamobjective. Tables4.3and4.4tabulatecombinatorialexchangesforthegiven initial allocations (X r 1 =∅,X r 2 ={x 1 ,x 2 },X r 3 ={x 3 ,x 4 }) with team cost 9. Task- allocation algorithms based on combinatorial exchanges actually evaluate all possible task allocations and thus consider all synergies among tasks. For example, Tables 4.3 and4.4enumerateallpossibleallocationsoffourtargetstothreerobots. Sinceallpossi- bletaskallocationsareevaluated,combinatorialexchangesproduceoptimalallocations. 2 There are two scenarios of how a task exchange is constructed between robots r and r ′ : i) robot r acts as the initiator and robot r ′ acts as the participant, and ii) robot r ′ acts as the initiator and robot r acts as the participant. Table 4.2 shows only one scenario for each single-task swap since the resulting allocationsareidenticalforbothscenarios. 100 I P1 P2 TasksfromI TasksfromP1 TasksfromP2 Newallocations Teamcost toP1 toP2 toI toP2 toI toP1 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ {x 3 ,x 4 } ∅ ({x 1 ,x 2 ,x 3 ,x 4 },∅,∅) 21 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ {x 3 } {x 4 } ({x 1 ,x 2 ,x 3 },{x 4 },∅) 21 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ {x 4 } {x 3 } ({x 1 ,x 2 ,x 4 },{x 3 },∅) 11 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ ∅ {x 3 ,x 4 } ({x 1 ,x 2 },{x 3 ,x 4 },∅) 11 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ {x 3 } ∅ ({x 1 ,x 2 ,x 3 },∅,{x 4 }) 21 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ {x 4 } ∅ ({x 1 ,x 2 ,x 4 },∅,{x 3 }) 9 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ ∅ {x 3 } ({x 1 ,x 2 },{x 3 },{x 4 }) 11 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ ∅ {x 4 } ({x 1 ,x 2 },{x 4 },{x 3 }) 7 r 1 r 2 r 3 ∅ ∅ {x 1 ,x 2 } ∅ ∅ ∅ ({x 1 ,x 2 },∅,{x 3 ,x 4 }) 9 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } {x 3 ,x 4 } ∅ ({x 1 ,x 3 ,x 4 },∅,{x 2 }) 11 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } {x 3 } {x 4 } ({x 1 ,x 3 },{x 4 },{x 2 }) 11 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } {x 4 } {x 3 } ({x 1 ,x 4 },{x 3 },{x 2 }) 11 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } ∅ {x 3 ,x 4 } ({x 1 },{x 3 ,x 4 },{x 2 }) 11 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } {x 3 } ∅ ({x 1 ,x 3 },∅,{x 2 ,x 4 }) 13 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } {x 4 } ∅ ({x 1 ,x 4 },∅,{x 3 ,x 4 }) 9 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } ∅ {x 3 } ({x 1 },{x 3 },{x 2 ,x 4 }) 13 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } ∅ {x 4 } ({x 1 },{x 4 },{x 2 ,x 3 }) 17 r 1 r 2 r 3 ∅ ∅ {x 1 } {x 2 } ∅ ∅ ({x 1 },∅,{x 2 ,x 3 ,x 4 }) 17 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } {x 3 ,x 4 } ∅ ({x 2 ,x 3 ,x 4 },∅,{x 1 }) 21 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } {x 3 } {x 4 } ({x 2 ,x 3 },{x 4 },{x 1 }) 21 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } {x 4 } {x 3 } ({x 2 ,x 4 },{x 3 },{x 1 }) 11 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } ∅ {x 3 ,x 4 } ({x 2 },{x 3 ,x 4 },{x 1 }) 11 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } {x 3 } ∅ ({x 2 ,x 3 },∅,{x 1 ,x 4 }) 21 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } {x 4 } ∅ ({x 2 ,x 4 },∅,{x 1 ,x 3 }) 9 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } ∅ {x 3 } ({x 2 },{x 3 },{x 1 ,x 4 }) 11 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } ∅ {x 4 } ({x 2 },{x 4 },{x 1 ,x 3 }) 11 r 1 r 2 r 3 ∅ ∅ {x 2 } {x 1 } ∅ ∅ ({x 2 },∅,,{x 1 ,x 3 ,x 4 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } {x 3 ,x 4 } ∅ ({x 3 ,x 4 },∅,{x 1 ,x 2 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } {x 3 } {x 4 } ({x 3 },{x 4 },{x 1 ,x 2 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } {x 4 } {x 3 } ({x 4 },{x 3 },{x 1 ,x 2 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } ∅ {x 3 ,x 4 } (∅,{x 3 ,x 4 },{x 1 ,x 2 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } {x 3 } ∅ ({x 3 },∅,{x 1 ,x 2 ,x 4 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } {x 4 } ∅ ({x 4 },∅,{x 1 ,x 2 ,x 3 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } ∅ {x 3 } (∅,{x 3 },{x 1 ,x 2 ,x 4 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } ∅ {x 4 } (∅,{x 4 },{x 1 ,x 2 ,x 3 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 ,x 2 } ∅ ∅ (∅,∅,{x 1 ,x 2 ,x 3 ,x 4 }) 17 (I=Initiator,P1=Participant1andP2=Participant2) Table4.3: CombinatorialExchangesforExampleinFigure4.1(MiniMaxTeamObjec- tive)(Part1) However,thetotalnumberofallocationsoftaskstorobotsconsideredinacombinatorial exchange is exponential in the number of tasks, which makes the task-allocation algo- rithm inefficient. Another issue of combinatorial exchanges is that it is unknown how to implement combinatorial exchanges in the framework of the multi-party negotiation protocol shown in Figure 1.10. As a result, Dias and Stentz’s approach uses a team leadertoconstructcombinatorialexchangesfortherobotteam. 101 I P1 P2 TasksfromI TasksfromP1 TasksfromP2 Newallocations Teamcost toP1 toP2 toI toP2 toI toP1 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ {x 3 ,x 4 } ∅ ({x 1 ,x 3 ,x 4 },{x 2 },∅) 9 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ {x 3 } {x 4 } ({x 1 ,x 3 },{x 2 ,x 4 },∅) 11 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ {x 4 } {x 3 } ({x 1 ,x 4 },{x 2 ,x 3 },∅) 17 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ ∅ {x 3 ,x 4 } ({x 1 },{x 2 ,x 3 ,x 4 },∅) 17 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ {x 3 } ∅ ({x 1 ,x 3 },{x 2 },{x 4 }) 9 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ {x 4 } ∅ ({x 1 ,x 4 },{x 2 },{x 3 }) 3 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ ∅ {x 3 } ({x 1 },{x 2 ,x 3 },{x 4 }) 17 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ ∅ {x 4 } ({x 1 },{x 2 ,x 4 },{x 3 }) 11 r 1 r 2 r 3 ∅ ∅ {x 1 } ∅ ∅ ∅ ({x 1 },{x 2 },{x 3 ,x 4 }) 9 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ {x 3 ,x 4 } ∅ ({x 2 ,x 3 ,x 4 },{x 1 },∅) 21 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ {x 3 } {x 4 } ({x 2 ,x 3 },{x 1 ,x 4 },∅) 21 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ {x 4 } {x 3 } ({x 2 ,x 4 },{x 1 ,x 3 },∅) 13 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ ∅ {x 3 ,x 4 } ({x 2 },{x 1 ,x 3 ,x 4 },∅) 13 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ {x 3 } ∅ ({x 2 ,x 3 },{x 1 },{x 4 }) 21 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ {x 4 } ∅ ({x 2 ,x 4 },{x 1 },{x 3 }) 9 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ ∅ {x 3 } ({x 2 },{x 1 ,x 3 },{x 4 }) 13 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ ∅ {x 4 } ({x 2 },{x 1 ,x 4 },{x 3 }) 7 r 1 r 2 r 3 ∅ ∅ {x 2 } ∅ ∅ ∅ ({x 2 },{x 1 },{x 3 ,x 4 }) 9 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } {x 3 ,x 4 } ∅ ({x 3 ,x 4 },{x 2 },{x 1 }) 7 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } {x 3 } {x 4 } ({x 3 },{x 2 ,x 4 },{x 1 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } {x 4 } {x 3 } ({x 4 },{x 2 ,x 3 },{x 1 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } ∅ {x 3 ,x 4 } (∅,{x 2 ,x 3 ,x 4 },{x 1 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } {x 3 } ∅ ({x 3 },{x 2 },{x 1 ,x 4 }) 7 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } {x 4 } ∅ ({x 4 },{x 2 },{x 1 ,x 3 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } ∅ {x 3 } (∅,{x 2 ,x 3 },{x 1 ,x 4 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } ∅ {x 4 } (∅,{x 2 ,x 4 },{x 1 ,x 3 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 1 } ∅ ∅ (∅,{x 2 },{x 1 ,x 3 ,x 4 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } {x 3 ,x 4 } ∅ ({x 3 ,x 4 },{x 1 },{x 2 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } {x 3 } {x 4 } ({x 3 },{x 1 ,x 4 },{x 2 }) 11 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } {x 4 } {x 3 } ({x 4 },{x 1 ,x 3 },{x 2 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } ∅ {x 3 ,x 4 } (∅,{x 1 ,x 3 ,x 4 },{x 2 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } {x 3 } ∅ ({x 3 },{x 1 },{x 2 ,x 4 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } {x 4 } ∅ ({x 4 },{x 1 },{x 2 ,x 3 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } ∅ {x 3 } (∅,{x 1 ,x 3 },{x 2 ,x 4 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } ∅ {x 4 } (∅,{x 1 ,x 4 },{x 2 ,x 3 }) 13 r 1 r 2 r 3 ∅ ∅ ∅ {x 2 } ∅ ∅ (∅,{x 1 },{x 2 ,x 3 ,x 4 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ ∅ {x 3 ,x 4 } ∅ ({x 3 ,x 4 },{x 1 ,x 2 },∅) 9 r 1 r 2 r 3 ∅ ∅ ∅ ∅ {x 3 } {x 4 } ({x 3 },{x 1 ,x 2 ,x 4 },∅) 11 r 1 r 2 r 3 ∅ ∅ ∅ ∅ {x 4 } {x 3 } ({x 4 },{x 1 ,x 2 ,x 3 },∅) 17 r 1 r 2 r 3 ∅ ∅ ∅ ∅ ∅ {x 3 ,x 4 } (∅,{x 1 ,x 2 ,x 3 ,x 4 },∅) 17 r 1 r 2 r 3 ∅ ∅ ∅ ∅ {x 3 } ∅ ({x 3 },{x 1 ,x 2 },{x 4 }) 9 r 1 r 2 r 3 ∅ ∅ ∅ ∅ {x 4 } ∅ ({x 4 },{x 1 ,x 2 },{x 3 }) 9 r 1 r 2 r 3 ∅ ∅ ∅ ∅ ∅ {x 3 } (∅,{x 1 ,x 2 ,x 3 },{x 4 }) 17 r 1 r 2 r 3 ∅ ∅ ∅ ∅ ∅ {x 4 } (∅,{x 1 ,x 2 ,x 4 },{x 3 }) 11 (I=Initiator,P1=Participant1andP2=Participant2) Table4.4: CombinatorialExchangesforExampleinFigure4.1(MiniMaxTeamObjec- tive)(Part2) 102 Sandholm et al. [TRS04, San98, AS99] study theoretical properties of multi-robot negotiationcontractsforre-allocatingtasks. Theyintroducethefollowingthreecontract typesthatarenecessarytoproduceoptimalallocationsfromanygiveninitialallocations, whereacontracttypecharacterizescertaintaskexchangesamongrobots: • Swap Contracts: A swap contract is the same as a task exchange with two non- emptytasks,inwhichonerobotrmovesasingletasktoanotherrobotr ′ androbot r ′ movesadifferenttasktorobotr atthesametime. • Cluster Contracts: In a cluster contract, one robot moves a bundle of tasks to anotherrobotatthesametime. • Multi-RobotContracts: A multi-robot contract involves at least three robots, in whicheachrobotmovesexactlyonetasktoanotherrobotatthesametime. 4.3 SequentialNegotiationswithK-Swaps In this section, we first propose a new contract type, called k-swaps, that characterizes a given number of task exchanges among multiple robots at the same time. After that, we show that k-swaps generalize all existing contract types discussed in the previous section. Finally,wedevelopanewnegotiationalgorithmforsolvingsimple-taskalloca- tionproblems,calledsequentialnegotiationswithK-swaps,inwhichrobotsrepeatedly perform a given number of task exchanges so that the current team cost decreases the mostineachround. 4.3.1 Overview Recall that we developed sequential auctions with bundles in Chapter 3 by exploring thesimilarityanddifferencebetweensequentialsingle-itemauctionsandcombinatorial 103 Single Task Exchanges Task Exchanges per Round Synergies Considered all limited k-Swaps k Combinatorial Exchanges k = 1 k = ∞ the more the larger k is Figure4.3: Connectionbetweenk-SwapsandExistingNegotiationAlgorithms auctions: Sequentialsingle-itemauctionsallocateonlyoneadditionaltasktosomerobot in each round, and combinatorial auctions allocate all tasks to one or more robots in a single round. Thus, sequential auctions with bundles generalize them by allocating k tasks to one or more robots in each round where 1≤ k <∞ is a given number, as showninFigure3.4. The same logic also applies to designing a new negotiation contract type: a single taskexchangedescribesexactlyonetaskexchangebetweentworobotsatthesametime, and combinatorial exchanges describe an arbitrary number of task exchanges among multiple robots (necessary to produce optimal allocations) at the same time. We there- foreproposeanewcontracttype,calledk-swaps,thatdescribesexactlyktaskexchanges among robots at the same time, where 1 ≤ k < ∞ is a given number. Figure 4.3 shows the connection between k-swaps and combinatorial exchanges and single task exchanges. Thenewcontracttypek-swapspromisestohaveabetterchancetodecreasetheteam cost of the initial allocations since it considers more task exchanges among robots than single task exchanges when k > 1. However, the technical difficulty of implementing k-swapsisthatitisunknownhowrobotscanconstructk-swapsinadistributedfashion 104 by using the multi-party negotiation protocol shown in Figure 1.10. We address this issueinthefollowingsubsections. 4.3.2 Partialk-Swaps We now introduce partialk-swaps, the basic building block of constructingk-swaps in a distributed fashion. In this subsection, we first describe several concepts for partial k-swaps, then discuss several operations on partial k-swaps, and finally prove several propertiesofpartialk-swaps. Concepts Let(X r ) r∈R = (X r 1 ,...,X rm )bethegiveninitialallocations. Wedefinethefollowing basictaskmovementsbetweentworobots: • Out Swap: An out swap of robot r ∈ R describes a task movement where task x ∈ X r is moved from robot r to some other unspecified robot, written as (r,−,x,−). • InSwap: Aninswapofrobotr∈ R describesataskmovementwheretaskx ′ / ∈ X r ismovedfromsomeotherunspecifiedrobottorobotr,writtenas(r,−,−,x ′ ). • ExchangeSwap: Anexchangeswapdescribestaskmovementsbetweentwodif- ferent robots r,r ′ ∈ R where task x∈ X r is moved from robot r to robot r ′ and task x ′ ∈ X r ′ is moved from robot r ′ to robot r, written as (r,r ′ ,x,x ′ ) (or, equivalently, (r ′ ,r,x ′ ,x)). At most one of the tasks x and x ′ in an exchange swap can be empty, written as ∅. A pair of exchange swaps (r,r ′ ,x,∅) and (r,r ′ ,∅,x ′ ) can be re-written as a single exchange swap (r,r ′ ,x,x ′ ). A set of exchange swaps is compact iff it does not contain pairsofexchangeswapsthatcanbere-writtenassingleexchangeswaps. 105 x 1 r 2 {x 1, x 2} OutSwap (r 2 ,−,x 1 ,−) x 3 r 2 {x 1 , x 2 } InSwap (r 2 ,−,−,x 3 ) r 2 x 3 {x 1, x 2 } x 1 r 3 {x 3 , x 4 } ExchangeSwap (r 2 ,r 3 ,x 1 ,x 3 ) Figure4.4: OutSwap,InSwapandExchangeSwapforExampleinFigure4.1 out swaps: (r 2 , -, x 1 , -) in swaps: (r 2 , -, -, x 3 ) exchange swaps: (r 2 , r 1 , x 2 , ϕ) r 2 {x 1 , x 2 } x 1 x 2 x 3 r 1 {} PartialOneSwap(s 1 ) netloss(r 1 ,s 1 ) =−2 netloss(r 2 ,s 1 ) = 2 loss(s 1 ) = 2 out swaps: in swaps: exchange swaps: (r 2 , r 1 , x 2 , ϕ) (r 2 , r 3 , x 1 , x 3 ) r 2 {x 1 , x 2 } x 1 x 2 x 3 r 1 {} r 3 {x 3 , x 4 } CompleteTwoSwap(s 2 ) netloss(r 1 ,s 2 ) =−2 netloss(r 2 ,s 2 ) = 2 netloss(r 3 ,s 2 ) =−2 loss(s 2 ) = 2 Figure 4.5: Partial and Complete k-Swap for Example in Figure 4.1 (MiniMax Team Objective) Consider again the multi-robot routing problem shown in Figure 4.1 for the Mini- Max team objective. Given the initial allocations (X r 1 =∅,X r 2 ={x 1 ,x 2 },X r 3 = {x 3 ,x 4 }), the left part of Figure 4.4 shows an example of an out swap, (r 2 ,−,x 1 ,−), which moves target x 1 from robot r 2 to some other unspecified robot. The bundle of targets{x 1 ,x 2 }isthecurrentallocationofrobotr 2 . ThecenterpartofFigure4.4shows an example of an in swap, (r 2 ,−,−,x 3 ), which moves targetx 3 to robotr 2 from some other unspecified robot. Finally, the right part of Figure 4.4 shows an example of an exchange swap, (r 2 ,r 3 ,x 1 ,x 3 ), which moves target x 1 from robot r 2 to robot r 3 and targetx 3 fromrobotr 3 torobotr 2 . Apartialk-swaps k describesthetaskmovementsamongasetofrobotsR(s k )⊆R. ItconsistsofasetofoutswapsinwhichtasksaremovedfromrobotsinR(s k )torobots not in R(s k ), a set of in swaps in which tasks are moved from robots not in R(s k ) to robots in R(s k ), and a compact set of exchange swaps in which tasks are moved 106 between robots that are in R(s k ). Each robot r∈ R(s k ) must appear at least once in an out, in or exchange swap of s k . The value k is the number of exchange swaps in s k . We sometimes refer to a partial k-swap as a partial multi-swap if the value of k is unspecified. The left part of Figure 4.5 shows an example of a partial one-swap that involves robots r 1 and r 2 for the multi-robot routing example shown in Figure 4.1 for theMiniMax teamobjective. This partialone-swapcontainsan outswap (r 2 ,−,x 1 ,−) thatmovestargetx 1 fromrobotr 2 tosomeunspecifiedrobotthatisnotinvolvedinthis partial one-swap, an in swap (r 2 ,−,−,x 3 ) that moves target x 3 to robot r 2 from some unspecified robot that is not involved in this partial one-swap, and an exchange swap (r 2 ,r 1 ,x 2 ,∅)thatmovestargetx 2 fromrobotr 2 torobotr 1 . A partial k-swap s k is complete iff its sets of out and in swaps are both empty. A complete (partial) k-swap describes exactly k task exchanges among multiple robots at the same time. Thus, complete k-swaps generalize both combinatorial exchanges (where k is at most the number of tasks) and single task exchanges (where k = 1). Complete k-swaps also generalize the three contract types introduced by Sandholm et al. [TRS04,San98,AS99],asfollows: • Swap Contracts: A swap contract is a complete one-swap with two non-empty tasksinitsexchangeswap. • ClusterContracts: Aclustercontractisacompletek-swapwithonlytworobots r andr ′ whose exchange swapsare of the form (r,r ′ ,x,∅) with different tasksx, wherek isthenumberoftasksmovedfromrobotr torobotr ′ . • Multi-RobotContracts: Amulti-robotcontractcanberepresentedasacomplete k-swapwithk robotsfork≥ 3. TherightpartofFigure4.5showsanexampleofacompletetwo-swapthatinvolves robotsr 1 ,r 2 andr 3 . Itcontainstwoexchangeswapsbutnooutorinswaps. 107 Let c be the current team cost of the given initial allocations (X r ) r∈R = (X r 1 ,...,X rm )beforeperforming(thetaskmovementsasspecifiedinapartialk-swap) s k . Let( X r ) r∈R = ( X r 1 ,... X rm )bethenewallocationsafterperformings k . Wedefine thenetlossofeachrobotr∈R(s k )tobe: netloss(r,s k ) :=    c robot r ( X r )−c robot r (X r ) (MiniSumandMiniLat) c robot r ( X r )−c. (MiniMax) For the MiniSum and MiniLat team objectives, netloss(r,s k ) is the value by which robot r increases its current robot cost after performing s k . For the MiniMax team objective, netloss(r,s k ) is the difference between the robot cost of robot r after per- forming s k and the team cost before performing s k . Consider again the multi-robot routingexampleshowninFigure4.1fortheMiniMaxteamobjective. Figure4.5shows the netloss of each robot involved in the partial one-swap and the complete two-swap. Forexample,thenetlossofrobotr 1 inthepartialone-swapis7-9=-2sincethecurrent team cost is 9 and the robot cost of robot r 1 for visiting target x 2 after performing this partialone-swapis 7. Now,wedefinethelossofapartialk-swaps k tobe: loss(s k ) =c team r∈R(s k ) netloss(r,s k ). For the MiniSum and MiniLat team objectives, theloss(s k ) = ∑ r∈R(s k ) netloss(r,s k ) is the total value by which robots in R(s k ) increase their robot costs after performing s k . For the MiniMax team objective, the loss(s k ) = max r∈R(s k ) netloss(r,s k ) is the difference between the largest robot cost of any robot inR(s k ) after performings k and 108 the team cost before performing s k . Figure 4.5 shows the losses of the partial one- swap and the complete two-swap for the example in Figure 4.1 for the MiniMax team objective. Finally,wedefineapartialk-swaps k tobeprofitableiff:    loss(s k )< 0 (MiniSumandMiniLat) loss(s k )< 0and∃r∈R(s k )c robot r (X r ) =c. (MiniMax) For the MiniSum and MiniLat team objectives, the partial k-swap s k is profitable iff the robots in R(s k ) decrease the sum of their robot costs after performing s k . For the MiniMaxteamobjective,however,onehastochecktwoconditionsinordertodetermine whetherthepartialk-swaps k isprofitable: i)therobotcostsofallrobotsinR(s k )after performing s k are less than the team cost before performing s k ; and ii) there exists at least one robot in R(s k ) whose current robot cost is equal to the team cost before performings k . The following theorem shows the reason why we are interested in profitable com- pletek-swaps. Theorem4.3.1 Onlyacompletek-swapthatisprofitablecandecreasetheteamcostof theinitialallocations. Proof: We first consider the MiniSum and MiniLat team objectives. For any complete k-swap s k that decreases the current team cost, we have ∑ r∈R c robot r X r − 109 ∑ r∈R c robot r X r < 0. Since s k only moves tasks among robots in R(s k ), the robots in R\R(s k )donotchangetheirallocationsafterperformings k . Thus, loss(s k ) = ∑ r∈R(s k ) netloss(r,s k ) = ∑ r∈R(s k ) (c robot r X r −c robot r X r ) = ∑ r∈R(s k ) c robot r X r − ∑ r∈R(s k ) c robot r X r = ∑ r∈R(s k ) c robot r X r − ∑ r∈R(s k ) c robot r X r + ∑ r∈R\R(s k ) c robot r X r − ∑ r∈R\R(s k ) c robot r X r = ∑ r∈R c robot r X r − ∑ r∈R c robot r X r < 0. We now consider the MiniMax team objective. For any complete k-swap s k that decreases the current team cost, we have max r∈R c robot r X r < c, which implies that c robot r ( X r )−c< 0foranyrobotr∈Randthusloss(s k ) = max r∈R(s k ) (c robot r ( X r )−c)< 0. This satisfies the first condition of s k being profitable. Now we consider the sec- ond condition: The completek-swaps k re-allocates tasks only among robots inR(s k ). Therefore,therobotcostsofrobotsinR\R(s k )donotchangeafterperformings k and thus are all less than the current team costc. There must be at least one robot inR(s k ) whosecurrentrobotcostisequaltoc. For the MiniSum and MiniLat team objectives, performing a profitable complete k-swap always decreases the team cost. For the MiniMax team objective, however, performingaprofitablecompletek-swaps k isnotguaranteedtodecreasetheteamcost sincetheremayberobotsinR\R(s k )whosecurrentrobotcostsareequaltothecurrent teamcostc. A partial k-swap s k is connected iff the graph is connected whose vertices are the robots in R(s k ) and whose edges connect two vertices iff they represent robots that appear in an exchange swap in s k . A disconnected partial k-swap s k can be viewed as 110 remove robot r 2 singleton swap: k r s 2 partial multi-swaps: k r k s s 2 / r 2 {x 1, x 2 } x 1 x 2 x 3 r 1 r 3 {x 3, x 4 } partial k-swap: k s r 2 {x 1, x 2 } x 1 x 3 x 2 x 1 x 3 r 3 {x 3 , x 4 } x 2 r 1 {} {} Figure4.6: RemovingRobotfromPartialk-Swap a set of two or more connected partial multi-swaps. In the following, all partial multi- swapsareassumedtobeconnectedunlessmentionedotherwise. Operations Wenowdefineoperationsonpartialk-swaps. An exchange swap (r,r ′ ,x,x ′ ) can be decomposed into an in swap (r,−,−,x ′ ) and an out swap (r,−,x,−) for robot r and an in swap (r ′ ,−,−,x) and an out swap (r ′ ,−,x ′ ,−) for robotr ′ . A robotr∈ R(s k ) can be removed from a partialk-swaps k asfollows: First,onedecomposesallexchangeswapsins k thatcontainrobotrandthen removes all out swaps and in swaps that contain robotr froms k . These out swaps and in swaps form a partial zero-swap that contains only robotr, called the singleton swap s k r . After removing robotr froms k , the remaining part ofs k is a set of connected par- tial multi-swaps, denoted by s k /s k r . Figure 4.6 shows one example of removing robot r 2 from a partial k-swap s k , which results in one singleton swap s k r 2 and two partial multi-swapss k /s k r 2 . Conversely, an in swap (r,−,−,x) and an out swap (r ′ ,−,x,−) can be combined to an exchange swap (r,r ′ ,∅,x). Such a pair of an out swap and an in swap forms a resolvablepair. A completek ′ -swaps k ′ completes a partialk-swaps k iff it results from adding out and in swaps to s k so that all out and in swaps can be grouped into resolvable pairs, combining each resolvable pair into an exchange swap and making the resulting set of 111 exchange swaps compact. A partialk-swaps k is bounded byK iff there is a complete k ′ -swaps k ′ withk ′ ≤ K that completes it. A partialk-swap is always bounded by the total number of its in, out, and exchange swaps although this bound is not necessarily tight. A partial g-swap s g and a partial h-swap s h are combinable iff they satisfy the fol- lowingconditions: • R(s g )∩R(s h ) =∅. • For each in swap (r,−,−,x) in s g or s h with x ∈ X r ′ for some robot r ′ ∈ R(s g )∪R(s h ), there must be an out swap (r ′ ,−,x,−) in s g or s h (that forms a resolvablepairwithit). • Theremustbeatleastoneresolvablepairins g ands h . The following operation combine(s g ,s h ) combines a combinable pair of a partial g-swaps g andapartialh-swaps h toanewpartialk-swaps k : 1. Addallexchangeswapsins g ands h tothesetofexchangeswapsins k . 2. For each resolvable pair of an in swap (r,−,−,x) and an out swap (r ′ ,−,x,−) in s g and s h , add the exchange swap (r,r ′ ,∅,x) to the set of exchange swaps in s k . 3. Makethesetofexchangeswapsins k compact. 4. Add each in swap or out swap ins g ands h that is not part of a resolvable pair to thesetsofinswapsoroutswapsofs k ,respectively. Thenewpartialk-swaps k containsallexchangeswapsins g ands h andoneormore additionalexchangeswapsthatresultfromcombiningtheresolvablepairsofoutswaps andinswapsins g ands h . 112 Theorem4.3.2 If s k = combine(s g ,s h ) for a combinable pair of a partial g-swap s g and a partial h-swap s h , then s k has the following properties: 1) s k is connected if s g ands h are connected, 2)R(s k ) = R(s g )∪R(s h ), 3)loss(s k ) = c team s∈{s g ,s h } loss(s), and 4)k≥g +h+1. Proof: 1) Since the multi-swapss g ands h are combinable, there exists at least one resolvable pair ins g ands h . The operationcombine(s g ,s h ) constructs a new exchange swap for any resolvable pair, and thus also connects s g and s h . 2) The operation combine(s g ,s h ) does not remove any robot from s g and s h , or add any robot to s g ands h . Thus,thenewpartialk-swaps k containsexactlyallrobotsins g ands h ,namely, R(s k ) =R(s g )∪R(s h ). 3)FortheMiniSumandMiniLatteamobjectives,itholdsthat loss(s k ) =loss(s g )+loss(s h )since: loss(s k ) = ∑ r∈R(s k ) (c robot r X r −c robot r X r ) = ∑ r∈R(s g )∪R(s h ) (c robot r X r −c robot r X r ) (R(s k ) =R(s g )∪R(s h )) = ∑ r∈R(s g ) (c robot r X r −c robot r X r ) + ∑ r∈R(s h ) (c robot r X r −c robot r X r ) (R(s g )∩R(s h ) =∅) = loss(s g )+loss(s h ). FortheMiniMaxteamobjective,itholdsthatloss(s k ) = max(loss(s g ),loss(s h ))since: loss(s k ) = max r∈R(s k ) (c robot r X r −c) = max r∈R(s g )∪R(s h ) (c robot r X r −c) (R(s k ) =R(s g )∪R(s h )) = max(max r∈R(s g ) (c robot r X r −c),max r∈R(s h ) (c robot r X r −c)) = max(loss(s g ),loss(s h )). 4)Theoperationcombine(s g ,s h )addsatleastoneexchangeswapsincethereexists at least one resolvable pair in the multi-swaps s g and s h . Any existing exchange swap 113 froms g (ors h ) involve both robots froms g (ors h ), and any new exchange swap added by the operation combine(s g ,s h ) involves one robot from s g and the other robot from s h . Therefore, existing exchange swaps from s g and s h do not change when the oper- ation combine(s g ,s h ) adds new exchange swaps and makes all exchange swaps in s k compact. Thus,wehavek≥g +h+1. Thefollowingtheoremshowshowrobotscanconstructpartialmulti-swaps. Theorem4.3.3 For any robotr∈ R(s k ) in a connected partialk-swaps k , if there are t connected partial multi-swaps in s k /s k r , then robot r can construct s k by using the combine operationttimestocombines k r withallpartialmulti-swapsins k /s k r . Proof: The partial multi-swaps in s k /s k r have the following two properties: i) they are all combinable with the singleton swap s k r since they are generated by removing robot r from the partial k-swap s k ; and ii) they do not share any robot since they are disconnected from each other. Let s ′ be initialized to s k r and (s 1 ,...,s t ) be any sequence of the t partial multi-swaps in s k /s k r . Robot r runs the following procedure: for i = 1,...t do s ′ = combine(s ′ ,s i ). The partial multi-swaps s ′ and s i are combin- ablesinceR(s ′ )∩R(s i ) =∅andallpreviouscombineoperations(ifany)donotchange anyexistingresolvablepairins k r ands i . Aftertheterminationofthisprocedure,alldis- connected partial multi-swaps in s k /s k r are connected back to s k r and thus the resulting partialmulti-swaps ′ isidenticaltotheoriginalpartialk-swaps k . Properties Wenowstudypropertiesofpartialmulti-swaps. Theorem4.3.4 Foranysimple task-allocationproblemwithntasksand anygiven ini- tial allocations (X r ) r∈R for this problem, there is always a complete n ′ -swap s n ′ with n ′ ≤nthattransforms (X r ) r∈R totheoptimalallocations. 114 … … … … r 1 r 2 r 3 x 1 x 2 x 3 r n r n-1 x n x n-1 x n-2 r n+1 Figure4.7: Example7ofMulti-RobotRouting Proof: Let the optimal allocations be ( ^ X r ) r∈R . We construct s n ′ as follows: s n ′ is empty initially. For any taskx∈ X withx∈ X r ∩ ^ X r ′ andr̸= r ′ , add the exchange- swap (r,r ′ ,x,∅), which moves task x from robot r to robot r ′ , to s n ′ . There are at mostn such exchange swaps, and this number may be even smaller after making these exchangeswapscompact. Theorem4.3.5 There exist simple task-allocation problems withn tasks and their ini- tialnon-optimalallocations(X r ) r∈R ,suchthattherearenoprofitablecompletek-swaps withk <n. Proof: Consider the multi-robot routing problem with n simple targets and known terrain shown in Figure 4.7. There exists only one capability. Robots r 1 ,...,r n+1 provide this capability and targets x 1 , ..., x n require this capability. The robot capacity of all n + 1 robots is 1. The optimal allocations of this prob- lem are ( ^ X r 1 , ^ X r 2 ,..., ^ X rn , ^ X r n+1 ) = ({x 1 },{x 2 },...,{x n },∅) with the smallest team cost 2n + 1 for the MiniSum and MiniLat team objectives and 3 for the Min- iMax team objective. Let (X r 1 ,X r 2 ,...,X rn ,X r n+1 ) = (∅,{x 1 },...,{x n−1 },{x n }) be the initial allocations with team cost 2(n + 1) for the MiniSum and Mini- Lat team objectives and 4 for the MiniMax team objective. The complete n-swap {(r 1 ,r 2 ,∅,x 1 ),...,(r i ,r i+1 ,∅,x i ),...,(r n ,r n+1 ,∅,x n )} is profitable and transforms the given initial allocations to the optimal ones. However, there does not exist any profitablecompletek-swapwithk <nforthisexample. 115 Theorem4.3.6 LetG = (V,E) be a connected graph with vertex weightsw v : v∈ V. Suppose ∑ v∈V w v = W < 0. Then existst∈ V such that, for every connected compo- nentC = (V C ,E C )ofG−t,thesumofthevertexweightsinC satisfies ∑ v∈V C w v < 0. Proof: For|V| ≤ 3 the theorem is clearly true. For our induction, let G be a counterexamplewith|V|minimum. LetS⊂V bethesetofverticesv suchthatG−v contains two or more connected components. (”S” stands for ”separator”). If S = ϕ selectt = argmax v∈V w v . Then ∑ v∈V\{t} w v < 0, G−t is connected, and hencet is asdesired. HenceforthS̸=ϕ. We claim that∃d∈ S and D ⊂ V such that D is the vertex set of a connected component of G−d and D ∩ S = ϕ. The proof of the claim is by induction on|V|. Chooses∈S arbitrarily. IfeveryconnectedcomponentofG−scontainsamemberof S,chooseoneconnectedcomponentG ′ = (V ′ ,E ′ ). Sinces∈S,|V ′ |<|V|−1. Joins withG ′ (usingalledgesinE betweensandV ′ )andcalltheresultinggraphG ′ +s. Ithas fewerthan|V| vertices. By induction it contains a separatord suchthatG ′ +s−d has a connected component D containing no separators of G ′ +s. Since G ′ is connected, d̸= s. Also, any separator of G in V ′ must be a separator of G ′ +s (since that is a subgraphofG). Thisprovestheclaim. Now let D,d be as must exist according to the claim. Let v∈ D. Since v̸∈ S, G−v is connected. Therefore w v ≤ W. (This is because G is a counterexample. If w v > W we could pick t = w.) Therefore ∑ v∈D w v ≤ ∑ v∈D W < 0. Since G is a counterexample,G−dcontainsaconnectedcomponentC withtotalvertexweight≥ 0. So C̸= D. Contract all vertices in D with d to get graph G dD with vertex dD having weightw d + ∑ v∈D w v . By construction, ∑ v∈G dD w v = W < 0 andG dD is connected. Byminimalityof|V|,inductivelyG dD containsavertextsatisfyingthestatementofthe theorem. However, it is not possible that dD = t because G−d contains a connected componentC̸= D with total weight≥ 0. Whent is removed fromG dD , vertexdD is 116 part of some connected component; expandingdD intod andD keeps the total weight of that component the same. Therefore t satisfies the conditions of the theorem for G. Theorem4.3.7 Let G = (V,E) be a connected graph with|V|≥ 2 and two weights per vertex,w 0 v andw 1 v : v∈ V. Supposew 0 v ≤ c∀v∈ V;w 1 v < c∀v∈ V, and∃u∈ V such thatw 0 u = c. Then∃t∈ V such thatt̸= u andG−t is connected. (Hence every connected component of G− t contains a vertex with w 0 = c, and all vertices have w 1 v <candw 0 v ≤c.) Proof: SinceG is connected a subset of its edges form a spanning treeT. T has at least 2 leaves, at least one of which, say t, is not u. Then T−t is a spanning tree on V\{t}, hence it is connected and contains u. Since G has all of T’s edges, G−t is connectedaswell. Corollary4.3.1 For any profitable connected partial k-swap s k , there is at least one robotr∈R(s k ),suchthatallconnectedpartialmulti-swapsins k /s k r areprofitable. Proof: LetG = (V,E) be the connected graph for the connected partialk-swaps k whose vertices are the robots in R(s k ) and whose edges connect two vertices iff they representrobotsthatappearinanexchangeswapins k . For the MiniSum and MiniLat team objectives, each vertex v ∈ V that maps to robot r∈ R(s k ) has a weight w v = loss(s k r ) : v∈ V. Since s k is profitable, then ∑ v∈V w v = ∑ r∈R(s k ) loss(s k r ) < 0. By Theorem 4.3.6, there exists t∈ V such that, for every connected componentC = (V C ,E C ) ofG−t, the sum of the vertex weights in C satisfies ∑ v∈V C w v < 0. Let robot r map to the vertex t, then each connected component C maps to a connected partial multi-swap in s k /s k r . Thus, all connected partialmulti-swapsins k /s k r areprofitableifweremoverobotr froms k . 117 For the MiniMax team objective, each vertex v∈ V that maps to robot r∈ R(s k ) has two weightsw 0 v = c robot r (X r ),w 1 v = c robot r ( X r ). Sinces k is profitable, thenw 0 v ≤ c and w 1 v < c for each vertex v∈ V. By Theorem 4.3.7, there exists t∈ V such that, every connected component C = (V C ,E C ) of G− t contains a vertex with w 0 = c, and all vertices have w 1 v < c and w 0 v ≤ c. Let robot r map to the vertex t, then each connected component C maps to a connected partial multi-swap in s k /s k r . Thus, all connectedpartialmulti-swapsins k /s k r areprofitableifweremoverobotr froms k . Assume that each robot r∈ R is allocated an index number index(r) so that all robots are ordered completely. Define a robot r ∈ R(s k ) to be the core of a partial k-swaps k iff the partial multi-swaps ins k /s k r are all profitable and there does not exist anyotherrobotr ′ ∈R(s k )withindex(r ′ )<index(r)thathasthisproperty. Corollary4.3.2 Foranyprofitablepartialk-swaps k ,thereisexactlyonecoreofs k . 4.3.3 NegotiationAlgorithm In this section, we develop a new negotiation algorithm, called sequential negotia- tions with K-swaps, in which robots construct and perform a complete k-swaps with 1≤k≤K (forauser-definedparameterK≥ 1)ineachroundsothatthecurrentteam cost decreases the most until the current team cost cannot be further decreased by any completek-swaps. Sequential negotiations withK-swaps consist of multiple rounds to re-allocatetasksamongrobots. Duringeachround,robotsconstructallprofitablepartial k-swaps with 1≤ k≤ K in the proposal-construction phase and select the most prof- itable complete one in the proposal-evaluation phase. The algorithm terminates when there are no profitable complete k-swaps with 1≤ k≤ K for the current allocations. Let the current allocations be (X r ) r∈R = (X r 1 ,...,X rm ). Now, we explain exactly whichproposalsrobotssendtoeachotherandhowtheyprocesstheseproposals: 118 • Task-Announcement Phase: Each robotr∈ R sends all other robots the infor- mationaboutitscurrentlyallocatedtasksX r ,itsindexnumberindex(r),and,for theMiniMaxteamobjective,itscurrentrobotcostc robot r (X r ). • Initial Task-Exchange Evaluation Phase: Each robotr∈ R first initializes the following sets to empty: the set T profitable r of profitable complete k-swaps with 1≤ k≤ K, the set S local r of partial multi-swaps that it has constructed, the set S send r ofpartialmulti-swapsthatitwillsendtoallotherrobots,andthesetS receive r of partial multi-swaps that it has received from other robots. After that, robot r constructs all possible partial zero-swaps bounded by K that contain only itself, adds them to S local r and, if they are profitable, also to S send r . Finally, it sends all partial zero-swaps in S send r as its proposals to all other robots and sets S send r to empty. • Proposal-Construction Phase: Each robot r∈ R repeats the following proce- dureforK iterations: 1. Each robot r adds each partial multi-swap that it receives from the other robotstoS receive r . 2. Eachrobotrcombineseverycombinablepairofpartialg-swaps g ∈S receive r and partial h-swap s h ∈ S local r as long as robot r is involved in at least one resolvable pair of s g and s h and executes for the resulting partial k-swap s k =combine(s g ,s h ): – If s k is a profitable complete k-swap bounded by K and robot r is the coreofs k ,thenrobotr addss k tothesetT profitable r . – Ifs k is not a completek-swap but bounded byK ands k / ∈ S local r , then robotr addss k toS local r and,ifs k isprofitableandrobotr isthecoreof s k ,alsotoS send r . 119 3. Each robotr sends all partial multi-swaps inS send r as its updated proposals toallotherrobotsandsetsS send r toempty. • Proposal-Evaluation Phase: Each robot r ∈ R sends all profitable complete multi-swapsinT profitable r toallotherrobotsandthenaddsallprofitablecomplete multi-swapsreceivedfromotherrobotstoT profitable r . Foreachcompletek-swaps s k ∈T profitable r ,eachrobotr∈Rcalculatesitsgainas gain(s k ) :=    −loss(s k ) (MiniSumandMiniLat) −max(loss(s k ),max r∈R\R(s k ) (c robot r (X r )−c)) (MiniMax) and determines the most profitable complete k ∗ -swaps s k ∗ = argmax s k ∈T profitable r gain(s k )(Allrobotsbreaktiesinthesameway). • Schedule-UpdatePhaseIfT profitable r ̸=∅,thenallrobotsperforms k ∗ onthecur- rentallocationsthatreducesthecurrentteamcostbygain(s k ∗ ),andthealgorithm startsanewroundtorepeatthisprocess. Otherwise,thealgorithmterminates. The following theorem proves that our algorithm constructs all profitable complete k-swaps with 1≤ k≤ K for the current allocations. Each profitable partialk-swap is sentbysomerobottoallotherrobotsatmostoncesinceitcanbesentonlybyitsunique corer. ThecorethenstoresitinS local r orT profitable r anddoesnotsenditagain. Theorem4.3.8 The core of any profitable partialk-swap bounded byK with 0≤ k≤ K has been constructed it by the end of the kth iteration of the proposal-construction stage. Proof: Weprovethetheorembyinductiononk. Itholdstriviallyfork = 0accord- ing to the description of the initial task-exchange evaluation stage. Assume that the 120 statement holds for all 0≤ k ′ < k. It then also holds for k: Every profitable partial k-swaps k hasauniquecorer∈R(s k )accordingtoCorollary4.3.2. Assumethatthere aretpartialmulti-swapsins k /s k r . Thesepartialmulti-swapsareallprofitableaccording to Corollary 4.3.1. Then, the following three properties hold: 1) t≥ 1 since k≥ 1 and there are thus at least two robots in R(s k ). 2) Each partial multi-swap in s k /s k r is boundedbyK sinces k isboundedbyK. 3)h≤k−t<k foreachpartialh-swaps h in s k /s k r because s k contains k exchange swaps and one needs to decompose at least one exchange swap for each one of the resulting t partial multi-swaps. Put together, each partial multi-swap ins k /s k r has been constructed by its core by the end of the (k−t)th iterationoftheproposalevaluationstageaccordingtotheinductionassumptionandwas thensenttoallotherrobots. Thus, robotr canconstructs k byusingthecombine oper- ation once in each one of the t iterations following the (k−t)th iteration to combine s k r with all partial multi-swaps in s k /s k r according to Theorem 4.3.3, which proves the theorem. Consider again the multi-robot routing example shown in Figure 4.1 for the Mini- Max team objective. Table 4.5 tabulates the profitable partial zero-swaps robots need to send to each other in the initial task-exchange evaluation stage of the first round of our negotiation algorithm for this example, which consist of a small part of all partial zero-swapsconstructedbyrobots. SequentialnegotiationswithK-swapsperformafinitenumber,lessthanorequalto K,oftaskexchangesamongrobotsineachroundsothatthecurrentteamcostdecreases the most. In other words, sequential negotiations with K-swaps achieve the myopic optimalityperformanceprincipleonmultipletasks. 121 Robots BoundedbyOne BoundedbyTwo (OutSwaps,InSwaps) Loss (OutSwaps,InSwaps) Loss r 3 ({(r 3 ,−,x 3 ,−)}, ∅) -4 ({(r 3 ,−,x 3 ,−),(r 3 ,−,x 4 ,−)}, ∅) -9 r 3 ({(r 3 ,−,x 3 ,−)}, {(r 3 ,−,−,x 1 )}) -2 ({(r 3 ,−,x 3 ,−),(r 3 ,−,x 4 ,−)}, {(r 3 ,−,−,x 1 )}) -4 r 3 ———————————————— ({(r 3 ,−,x 3 ,−)}, {(r 3 ,−,−,x 2 )}) 4 ——————————————————————– ({(r 3 ,−,x 3 ,−),(r 3 ,−,x 4 ,−)}, {(r 3 ,−,−,x 2 )}) 2 r 3 ({(r 3 ,−,x 4 ,−)}, ∅) -6 ————————————————————————————— ({(r 3 ,−,x 3 ,−),(r 3 ,−,x 4 ,−)}, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 2 )}) 2 r 3 ———————————————— ({(r 3 ,−,x 4 ,−)}, {(r 3 ,−,−,x 1 )}) 2 ——————————————————————– ({(r 3 ,−,x 3 ,−)}, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 2 )}) 4 r 3 ———————————————— ({(r 3 ,−,x 4 ,−)}, {(r 3 ,−,−,x 2 )}) 8 ——————————————————————– ({(r 3 ,−,x 4 ,−)}, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 2 )}) 8 r 3 —————————– (∅, {(r 3 ,−,−,x 1 )}) 2 ———————————————— (∅, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 2 )}) 8 r 3 —————————– (∅, {(r 3 ,−,−,x 2 )}) 8 r 2 ({(r 2 ,−,x 1 ,−)}, ∅) -6 ({(r 2 ,−,x 1 ,−),(r 2 ,−,x 2 ,−)}, ∅) -9 r 2 ———————————————— ({(r 2 ,−,x 1 ,−)}, {(r 2 ,−,−,x 3 )}) 8 ——————————————————————– ({(r 2 ,−,x 1 ,−),(r 2 ,−,x 2 ,−)}, {(r 2 ,−,−,x 3 )}) 2 r 2 ———————————————— ({(r 2 ,−,x 1 ,−)}, {(r 2 ,−,−,x 4 )}) 2 ({(r 2 ,−,x 1 ,−),(r 2 ,−,x 2 ,−)}, {(r 2 ,−,−,x 4 )}) -4 r 2 ({(r 2 ,−,x 2 ,−)}, ∅) -4 ————————————————————————————— ({(r 2 ,−,x 1 ,−),(r 2 ,−,x 2 ,−)}, {(r 2 ,−,−,x 3 ),(r 2 ,−,−,x 4 )} 2 r 2 ———————————————— ({(r 2 ,−,x 2 ,−)}, {(r 2 ,−,−,x 3 )}) 4 ——————————————————————– ({(r 2 ,−,x 1 ,−)}, {(r 2 ,−,−,x 3 ),(r 2 ,−,−,x 4 )}) 8 r 2 ({(r 2 ,−,x 2 ,−)}, {(r 2 ,−,−,x 4 )}) -2 ——————————————————————– ({(r 2 ,−,x 2 ,−)}, {(r 2 ,−,−,x 3 ),(r 2 ,−,−,x 4 )}) 4 r 2 —————————– (∅, {(r 2 ,−,−,x 3 )}) 8 ———————————————— (∅, {(r 2 ,−,−,x 3 ),(r 2 ,−,−,x 4 )}) 8 r 2 —————————– (∅, {(r 2 ,−,−,x 4 )}) 2 r 1 —————————– (∅, {(r 1 ,−,−,x 1 )}) -8 ———————————————— (∅, {(r 1 ,−,−,x 1 ),(r 1 ,−,−,x 2 )}) -2 r 1 —————————– (∅, {(r 1 ,−,−,x 2 )}) -2 ———————————————— (∅, {(r 1 ,−,−,x 1 ),(r 1 ,−,−,x 3 )}) 0 r 1 —————————– (∅, {(r 1 ,−,−,x 3 )}) -2 ———————————————— (∅, {(r 1 ,−,−,x 1 ),(r 1 ,−,−,x 4 )}) -6 r 1 —————————– (∅, {(r 1 ,−,−,x 4 )}) -8 ———————————————— (∅, {(r 1 ,−,−,x 2 ),(r 1 ,−,−,x 3 )}) 12 r 1 ———————————————— (∅, {(r 1 ,−,−,x 2 ),(r 1 ,−,−,x 4 )}) 0 r 1 ———————————————— (∅, {(r 1 ,−,−,x 3 ),(r 1 ,−,−,x 4 )}) -2 Table 4.5: Partial Zero-Swaps Sent for Example in Figure 4.1 (MiniMax Team Objec- tive) 4.4 ExperimentalEvaluation Wenowevaluatethe performance of sequential negotiationswithK-swapsempirically forsolvingsimple-taskallocationproblemsinthedomainofmulti-robotrouting. 4.4.1 ExperimentalSettings Wecomparethefollowingsimple-taskallocationalgorithms: • NegotiationAlgorithms: – SNW (K = 1): This algorithm implements sequential negotiations with one-swaps. – SNW (K = 2): This algorithm implements sequential negotiations with two-swaps. 122 – SNW (K = 3): This algorithm implements sequential negotiations with three-swaps. • AuctionAlgorithms: – SAB (k = 1): This algorithm implements sequential auctions with bundle sizeone. – SAB (k = 2): This algorithm implements sequential auctions with bundle sizetwo. – SAB (k = 3): This algorithm implements sequential auctions with bundle sizethree. • CentralizedAlgorithms: – Optimal: Thisalgorithmimplementstheoptimalalgorithm. – Rollout: Thisalgorithmimplementsthecentralizedrolloutalgorithmaswe describedinSubsection3.4.1. – Random: This algorithm implements the randomized task-allocation algo- rithmaswedescribedinSubsection3.4.1. We use the same experimental settings from Subsection 3.4.1 to evaluate our task- allocationalgorithmsforsolvingmulti-robotroutingproblemswiththreedifferentkinds ofterrain,differentrobotcapacities,anddifferentapproachesofrobot-costcalculations. 4.4.2 ExperimentalResults We perform experiments with different settings introduced in the previous subsection to evaluate our simple-task negotiation algorithms, as summarized in Table 4.6. The 123 Results Terrain Blockage Capacities Robots Targets Optimal Robot-Cost Initial Instances Status Algorithm Calculation Allocations TableA.22 TeamCosts Outdoor-Like Known 3 2-10 6-30 Yes Approximated SAB(k =1) 15 TableA.23 TeamCosts Outdoor-Like Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TablesA.24andA.25 Communications Outdoor-Like Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.26 ResponseTimes Outdoor-Like Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.27 TeamCosts Empty Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.28 ResponseTimes Empty Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.29 TeamCosts Office-Like Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.30 ResponseTimes Office-Like Known 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.31 TeamCosts Outdoor-Like Known 2-5 10 20-50 No Approximated SAB(k =1) 100 TableA.32 ResponseTimes Outdoor-Like Known 2-5 10 20-50 No Approximated SAB(k =1) 100 TableA.33 TeamCosts Outdoor-Like Known 2-5 10 20-50 No Ideal SAB(k =1) 100 TableA.34 ResponseTimes Outdoor-Like Known 2-5 10 20-50 No Ideal SAB(k =1) 100 TableA.35 TeamCosts Outdoor-Like Known 3 2-10 6-30 No Approximated Random 100 TableA.36 ResponseTimes Outdoor-Like Known 3 2-10 6-30 No Approximated Random 100 TableA.37 TeamCosts Outdoor-Like Unknown 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.38 RepeatedExecutions Outdoor-Like Unknown 3 2-10 6-30 No Approximated SAB(k =1) 100 TableA.39 ResponseTimes Outdoor-Like Unknown 3 2-10 6-30 No Approximated SAB(k =1) 100 Table4.6: SummaryofSimple-TaskNegotiationExperiments detailed experimental data and explanations can be found in Appendex A.2. We make thefollowingobservations: TeamCosts • The team costs of sequential negotiations withK-swaps with a larger number of K are smaller than those of sequential negotiations withK-swaps with a smaller number of K. This observation meets our expectation since sequential negotia- tionswithK-swapswitha largernumberofK constructandperform moreprof- itable task exchanges than sequential negotiations with K-swaps with a smaller numberofK. • TheteamcostsofsequentialnegotiationswithK-swapsaresmallerthanthoseof theinitialallocations,nomatterhowtheinitialallocationsareproduced. • The team costs of sequential negotiations with K-swaps are different if the ini- tialallocationsareproducedbydifferenttask-allocationalgorithms,however,the differenceisinsignificantanddiminishesasthenumberofK increases. • The team costs of sequential negotiations with K-swaps are very close to the minimalonesforallthreeteamobjectiveswithasmallnumberofK. 124 Communication • The number of profitable partial (and complete) multi-swaps increases when the problemsizeincreases. Thisobservationmeetsourexpectationsincethenumber of possible task exchanges, and thus the number of the profitable ones, increases whentherearemorerobotsandtargets. • The number of profitable partial multi-swaps increases when the number of K in sequential negotiations with K-swaps increases. This observation meets our expectation since the number of possible task exchanges, and thus the number of theprofitableones,increaseswhenthenumberofK islarger. • The number of profitable partial multi-swaps (that robots need to send) is only a small part of the partial multi-swaps (that robots construct) in each round. For example, the number of profitable partial multi-swaps is consistently less than 5 percent of the number of partial multi-swaps in sequential negotiations with K swaps (K = 1, 2, 3) for all three team objectives when there are 10 robots and 30 targets. ResponseTimes • The response times of sequential negotiations with K-swaps increase with the numberofK. Thisobservationmeetsourexpectationsincethenumberofpartial multi-swapsincreasesexponentiallywiththenumberofK. • TheresponsetimesofsequentialnegotiationswithK-swapswhentheinitialallo- cationsare produced by the randomized algorithm are largerthan those when the initial allocations are produced by sequential single-item auctions. This observa- tion meets our expectation since the number of profitable partial multi-swaps is muchlargerwheninitialallocationsaremoresub-optimal. 125 • The response times of sequential negotiations withK-swaps with small numbers ofK (K =1or2)aresmall(lessthan5seconds). 4.5 Summary In this chapter, we studied negotiation algorithms for solving simple task-allocation problems. We developed a new negotiation algorithm, called sequential negotiations with K-swaps, that performs a given number of task exchanges among robots in each roundsothat theteam costdecreases themost. Our theoreticalresults showthatrobots need to send only profitable k-swaps to other robots sequential negotiations with K- swaps. OurexperimentalresultsshowthatsequentialnegotiationswithK-swapswitha smallnumberofK canreducetheteamcostsofthegiveninitialallocationstobeclose tothesmallestoneswithsmallresponsetimes. 126 Chapter5 Complex-TaskAuctions Inthischapter,westudytask-allocationalgorithmsthatimplementtheiterativeauction protocolshowninFigure1.5forsolvingcomplex-taskallocationproblems. Themajor- ityofexistingauctionalgorithms[TLJK05,DZZS04,Sar07]consideronlysimpletasks, whichcanbeexecutedbysinglerobots. However,therearecomplextasksinmanyreal- world applications [KEK11, ZP12], which need to be executed by more than one robot atthesametime. Forexample,robotshavetocooperatetomoveheavyrocksoutofthe road, and fire engines have to cooperate to extinguish large fires. Remember that one ofthemainmotivationsofusingrobotteamsisthatagroupofrobotscanexecutetasks that single robots cannot. We therefore believe that it is important to extend existing auction algorithms to be capable of allocating complex tasks to robots effectively and efficiently. Undoubtedly, the existence of complex tasks makes task-allocation problems more difficulttosolve. Atask-allocationalgorithmforallocatingcomplextaskstorobotshas tosolvethefollowingtwointerrelatedsubproblems: • Subproblem 1: Each complex task has to be executed by more than one robot. Thus,onehastodeterminewhichgroupofrobotsshouldexecuteagivencomplex task. • Subproblem 2: Each complex task has to be executed by a group of robots at the same time. Thus, one has to determine when a given group of robots should executeagivencomplextask. 127 Inthischapter,weillustratethatsequentialsingle-itemauctions[TLJK05],theexist- ingauctionalgorithmforallocatingsimpletaskstorobots,canbeextendedtoallocating both simple and complex tasks to robots. Our goal is to develop auction algorithms for solving complex-task allocation problems (complex-task auctions) that achieve the followingmyopicoptimizationperformanceprincipleonsingletasks: Performance principle of complex-task auctions: To allocate one addi- tionalcomplexorsimpletasktorobotsineachroundofcomplex-taskauc- tionssothatthecurrentteamcostincreasestheleast. We proceed as follows: Section 5.1 formalizes complex-task allocation problems. Although the existing approaches in the literature do not apply directly to solving complex-task allocation problems, Section 5.2 introduces a trivial extension of these approaches to allow them to allocate complex tasks to robots. We demonstrate that such an extension cannot achieve the performance principle of complex-task auctions. Section 5.3 then introduces a novel data structure, called reaction functions, that suc- cinctly characterizes the local information of robots needed by the auctioneer in order to achieve the performance principle of complex-task auctions. After that, we develop a new auction algorithm, called sequential auctions with reaction functions, in which robots construct and bid reaction functions and the auctioneer allocates one additional complex or simple task to robots in each round so that the current team cost increases the least. In order to better illustrate the concept and how reaction functions are used, we assume in Section 5.3 that robots can execute at most one complex task. Section 5.4 relaxes this assumption and generalizes the concept of reaction functions to multi- ple complex tasks. Finally, Section 5.5 presents experimental results, and Section 5.6 concludesthischapter. 128 5.1 ProblemDescription We now formalize complex-task allocation problems. A complex-task allocation prob- lem⟨R,X,E⟩ consists of a finite set of robots R ={r 1 ,...,r m }, a finite set of tasks X ={x 1 ,...,x n }, and a finite set of capabilitiesE ={e 1 ,...,e o }. Each robotr∈ R is characterized by a vector of capabilities E r = (e r 1 ,...,e r o ) that it provides, where each e r i is binary and ∑ o i=1 e r i = 1 (each robot provides one and only one capability). Each task x∈ X is characterized by a vector of capabilities E x = (e x 1 ,...,e x o ) that it requires, where each e x j is a non-negative integer and ∑ o j=1 e x j > 0. A robot r∈ R is qualified for task x∈ X iff E x ≥ E r . We define the coalition size d(x) of task x to be ∑ o j=1 e x j . We call a task x simple if d(x) = 1 and complex otherwise. 1 The set of simple tasks and the set of complex tasks partition the set of all tasks. We distinguish these two kinds of tasks because a robot can freely determine when to execute simple tasksbutneedstoagreewithotherrobotsonwhentoexecutecomplextasks. Acoalition of robotsP is qualified for taskx iff ∑ r∈P E r ≥ E x . For each complex taskx∈ X, a qualified coalitionofd(x)differentrobotsneedstoexecuteitatsomeexecutiontimet. Eachrobotinthatcoalitionthenhasacommitmenttoexecutetaskxatexecutiontimet, writtenasx←t. Eachrobotr∈ R isalso characterizedby a simple-task capacityq r s andacomplex- task capacity q r c , with the meaning that it can execute at most q r s simple tasks and at most q r c complex tasks. If q r c = 1 for all robots r∈ R in a complex-task allocation problem, then we call it one with disjoint coalitions since one robot cannot be part of more than one coalition. Otherwise, we call the complex-task allocation problem one with(potentially)overlappingcoalitions. 1 In this dissertation, we define each robot has only one capability in order to make sure that any complextaskhastoallocatetomorethanonerobot. 129 An allocation of robot r consists of a triple (X r s ,X r c ,C r ), where X r s is the set of simple tasks allocated to it,X r c is the set of complex tasks allocated to it, andC r is the setofitscommitmentsforexecutingthecomplextasksinX r c . Now we define the robot cost c robot r (X r s ,X r c ,C r ) of robot r for a given allocation (X r s ,X r c ,C r ). Given an execution order of tasks inX r s ∪X r c , the task-completion time of robot r is the largest execution time of any task in X r s ∪X r c if robot r can execute its complex tasks in X r c at the execution times specified in C r without waiting for any simple tasks (and infinity otherwise). For both the MiniSum and MiniMax team objec- tives, the robot cost c robot r (X r s ,X r c ,C r ) is the smallest task-completion time of robot r for executing its allocated tasks in X r s ∪ X r c in an optimal execution order. For the MiniLat team objective, we intend to minimize the sum of the execution times of all tasks. However, a complex task is allocated to more than one robot in a complex-task allocationproblem. Wethereforedefinethenormalizedlatencyofataskxtobeitsexe- cutiontimedividedbyitscoalitionsized(x). Then,foragivenexecutionorderoftasks in X r s ∪X r c , the cumulative latency of robot r is the sum of the normalized latencies of tasks in X r s ∪ X r c if it can execute its complex tasks in X r c at the specified execu- tion times in C r without waiting for any simple tasks (and infinity otherwise). For the MiniLat team objective, the robot cost is the smallest cumulative latency of robotr for executing its tasks inX r s ∪X r c in an optimal execution order. We define the robot cost c robot r (X r s ,X r c ,C r ) =∞if|X r s |>q r s or|X r c |>q r c . Finally, we want to to find a solution with a small team cost, where a solution of a complextaskallocationproblemrequireseachsimpletasktobeallocatedtoexactlyone qualified robot and each complex task to be allocated to exactly one qualified coalition ofrobotsandallrobotsinthecoalitionhavethesamecommitmentforit. Theteamcost c team r∈R c robot r (X r s ,X r c ,C r ) is ∑ r∈R c robot r (X r s ,X r c ,C r ) for both the MiniSum and MiniLat teamobjectivesand max r∈R c robot r (X r s ,X r c ,C r )fortheMiniMaxteamobjective. 130 5.2 ExistingApproachestoComplexTaskAuctions Although researchers begin to recognize the importance of allocating complex tasks to robotsinarobotsystem[GM04,BKL10,KEK11,ZP12],itisstillunknownhowauction algorithmscanallocatethemsothattheperformanceprincipleofcomplex-taskauctions isachieved. Sequentialsingle-itemauctions[TLJK05],asdiscussedinSubsection3.2.3,allocate simple tasks to robots in multiple rounds. In each round, each robot bids numerical values on single unallocated simple tasks, and the auctioneer selects the smallest bid as the winning bid and allocates one additional simple task to one robot. Sequential single-itemauctionsdonotapplydirectlytoallocatingcomplextaskstorobotsbecause robots cannot bid numerical values on complex tasks given that the cost of a robot for executinganadditionalcomplextaskdependsonitsexecutiontime. Research on allocating complex tasks is usually referred to as coalition forma- tion [KR84] in the literature. The coalition-formation algorithm most relevant for our complex-taskallocationproblemsisbyShehoryandKraus[SK95,SK98]. Thestructure of their algorithm is similar to sequential single-item auctions. It consists of multiple rounds to allocate complex tasks to coalitions of robots. In each round, each robot bids a numerical value on each unallocated complex task and the auctioneer evaluates all possibleallocationsofanunallocatedcomplextasktoaqualifiedcoalitionofrobotsand determines the winning bid with the smallest evaluationcost. In their algorithm, robots are able to bid numerical values on complex tasks since they assume that the cost of a robot for executing an additional complex task is solely determined by the robot and complex task. However, this assumption does not hold for our complex-task allocation problems since the cost of a robot for executing an additional complextask depends on itsexecutiontime. 131 Inordertosatisfythisassumptionforsolvingourcomplex-taskallocationproblems, one can restrict robots to execute their tasks in the same order in which the tasks are allocated to them, similar to the idea used in [SB05, VH07]. This way, each task is allocated with an unchangeable execution time, and robots always execute their tasks at the allocated execution times. Thus, the cost of a robot for executing an additional task can be uniquely determined since the robot can only execute that task after all previously allocated ones. With this restriction, we now extend sequential single-item auctions to a new auction algorithm, called SK auctions, that can solve complex-task allocationproblems. An SK auction allocates tasks to robots in multiple rounds. Initially, all tasks are unallocated. During each round, all robots bid numerical values on single unallocated tasks, and the auctioneer then allocates one additional task to robots with an unchange- able execution time. Consider any round of the SK auction. Assume that robot r∈ R has already been allocated the tasks X r = X r s ∪X r c ⊆ X in previous rounds. Thus, U = X\ ∪ r∈R X r is the set of unallocated tasks. Let t r be the execution time of the taskallocatedtorobotr withthelargestexecutiontime,whichisequaltotheexecution timeofthelasttaskx r intheexecutionorderofrobotr. Wenowexplainexactlywhich bidsrobotssubmitandhowtheauctioneermakestheallocationdecision. • Bid-Construction Phase: A robot r∈ R is eligible for executing an additional simple unallocated taskx s ∈ U iff|X s r | < q r s and robotr is qualified for taskx s . Similarly, robotr is eligible for executing an additional complex taskx c ∈ U iff |X c r |<q r c androbotr isqualifiedfortaskx c . For each unallocated task x∈ U, if robot r is still eligible for executing it, then it calculates the smallest execution timet r x of taskx if it executes taskx after all tasks in X r . For example, t r x = t r +d(x r ,x) in a multi-robot routing problem, 132 whered(x r ,x) is the travel time of robotr from the location of the last targetx r initsexecutionordertothelocationoftargetx. Then,robotr bidst r x ontaskx. • Winner-Determination Phase: Remember that solving a complex-task alloca- tion problem involves solving two subproblems: 1) which coalition of robots should execute a given task; and 2) when a given coalition of robots should exe- cuteagiventask. 2 – Solution of the first subproblem: For each unallocated task x ∈ U, the auctioneer first constructs the set P(d(x)) that contains all qualified coalitions of d(x) different eligible robots for executing task x. For each coalition P ∈ P(d(x)), the auctioneer then determines the optimal exe- cution time t P x for the robots in P to execute task x after their currently allocated tasks as explained below in the solution of the second subprob- lem. After that, the auctioneer determines the optimal pair (P,x) := argmin P∈P(d(x)),x∈U ∑ r∈P (t P x − t r ) for the MiniSum team objective and (P,x) = argmin P∈P(d(x)),x∈U t P x for both the MiniMax and MiniLat team objectives. Finally, the auctioneer allocates task x to all robots in P with executiontimet P x . – Solution of the second subproblem: For each pair of a qualified coalition of eligible robotsP∈ P(d(x)) and an unallocated taskx∈ U, the auction- eer determines t P x = max r∈P t r x to be the optimal execution time of task x giventhattaskxcanonlybeexecutedwhenallrobotsinP areavailablefor executingit. 2 We abuse the definition of coalitions here so that coalitions may consist of only single robots for executingsimpletasks. 133 r 1 x 1 x 4 r 2 x 2 x 5 x 3 Figure5.1: Example9ofMulti-RobotRouting r 1 x 1 x 4 r 2 x 2 x 5 x 3 OptimalAllocations (TeamCost=13) r 1 x 1 x 4 r 2 x 2 x 5 x 3 AllocationsofSKAuctions (TeamCost=14) Figure5.2: AllocationsforExampleinFigure5.1(MiniMaxTeamObjective) SKauctions,ourfirstattemptatsolvingcomplex-taskallocationproblems,canallo- cate complex tasks to robots quickly because the execution time of any complex task for a given coalition of robots can be trivially determined as the largest execution time submitted by the robots in that coalition. Simple tasks are allocated in the same way as complextasksinSKauctions. Figure 5.1 shows an example of multi-robot routing with complex targets in known terrain. It consists of two robots, four simple targets and one complex target. There exists only one capability. Robots r 1 and r 2 provide one unit of this capability each. Targetsx 1 ,x 2 ,x 3 andx 4 requireoneunitofthiscapabilityeach,whiletargetx 5 requires twounits. Thesimple-targetcapacityofbothrobotsis4,andthecomplex-targetcapac- ity of both robots is 1. The left part of Figure 5.2 shows optimal allocations with the smallest team cost 13 for the MiniMax team objective: First, robotr 2 visits targetx 3 at 134 BidConstruction WinnerDetermination CurrentAllocations Bids(calculatedandsubmitted) Decision Round1 - Robotr 1 bids: 3ontargetx 1 Theauctioneerallocatestargetx 3 3ontargetx 2 torobotr 2 atvisittime2. 7ontargetx 3 7ontargetx 4 2ontargetx 5 - Robotr 2 bids: 8ontargetx 1 8ontargetx 2 2ontargetx 3 2ontargetx 4 3ontargetx 5 Round2 - Robotr 1 bids: 3ontargetx 1 Theauctioneerallocatestargetx 1 3ontargetx 2 torobotr 1 atvisittime3. 7ontargetx 4 2ontargetx 5 Robotr 2 visitsx 3 attime2 Robotr 2 bids: 10ontargetx 1 12ontargetx 2 6ontargetx 4 7ontargetx 5 Round3 Robotr 1 visitsx 1 attime3 Robotr 1 bids: 9ontargetx 2 Theauctioneerallocatestargetx 4 13ontargetx 4 torobotr 2 atvisittime6. 8ontargetx 5 Robotr 2 visitsx 3 attime2 Robotr 1 bids: 12ontargetx 2 6ontargetx 4 7ontargetx 5 Round4 Robotr 1 visitsx 1 attime3 Robotr 1 bids: 9ontargetx 2 Theauctioneerallocatestargetx 2 8ontargetx 5 torobotr 1 atvisittime9. Robotr 2 visitsx 3 attime2 Robotr 2 bids: 14ontargetx 2 andthenx 4 attime6 11ontargetx 5 Round5 Robotr 1 visitsx 1 attime3 Robotr 1 bids: 14ontargetx 5 Theauctioneerallocatestargetx 5 andthenx 2 attime9 torobotsr 1 andr 2 atvisit Robotr 2 visitsx 3 attime2 Robotr 1 bids: 11ontargetx 5 time14. andthenx 4 attime6 Table5.1: SKAuctionsforExampleinFigure5.1(MiniMaxTeamObjective) time 2, and robot r 1 visits target x 1 at time 3. Then, robot r 2 arrives at complex target x 5 attime7andwaitsthereuntilrobotr 1 arrivesattime8. Finally,robotr 2 visitstarget x 4 attime13,androbotr 1 visitstargetx 2 attime13. Table5.1tabulatesthebidsrobots submit and the decisions of the auctioneer of SK auctions for this example. The right partofFigure5.2showstheresultingallocationsofSKauctionsfortheMiniMaxteam objective. SK auctions does not always allocate a task to robots so that the current team cost increasestheleast. Consideragainthemulti-robotroutingexampleshowninFigure5.1 for the MiniMax team objective. In the last round of SK auctions for this example, 135 as shown in Table 5.1, complex target x 5 is allocated to robots r 1 and r 2 at time 14. However,therobotscouldvisittargetx 5 attime8andvisittheirallocatedsimpletargets attheoptimaltimes,whichwouldreducetheteamcostto13. Thus,SKauctionsdonot achievetheperformanceprincipleofcomplex-taskauctions. 5.3 DisjointCoalitions The main drawback of SK auctions is that each task is allocated with an unchangeable execution time. Thus, it is natural to conjecture that one can produce better allocations ifrobotsareabletochangetheexecutiontimesoftheirallocatedtaskswhentheybidon unallocatedones. Inthissection,weintroduceanewauctionalgorithm,calledsequentialauctionswith reaction functions, in which robots can change the execution times of their allocated tasks and the auctioneer allocates one additional (simple or complex) task to robots in eachroundsothatthecurrentteamcostincreasestheleast. Inordertomakeiteasyfor readers to understand this algorithm, we start with the case of disjoint coalitions in this section, namely, where each robot can execute at most one complex task. We study the generalcaseofoverlappingcoalitionsinthenextsection. 5.3.1 Overview Wefirstconsiderhowtoallocateoneadditionalcomplextasktorobotssothatthecurrent teamcostincreasestheleast,andthendeveloptheauctionalgorithmthatallocatesboth simple and complex tasks to robots. In order to allocate a complex task to the optimal coalition of robots at the optimal execution time so that the current team cost increases theleast,theauctioneercouldconstructallpossiblequalifiedcoalitionsofeligiblerobots forexecutingthecomplextask(solutionofthefirstsubproblem)andthendeterminethe 136 optimal execution time of the complex task for each coalition of robots (solution of the second subproblem). The difficulty is in solving the second subproblem, namely, how to determine the optimal execution time of a given complex task for a given coalition of robots. SK auctions allocate tasks with unchangeable execution times so that the auctioneercaneasilydeterminetheexecutiontimeofthecomplextask. However,robots in the given coalition have to solve difficult scheduling and coordination problems for determining the optimal execution time of the given complex task when the execution timesoftheirallocatedtaskscanbechanged. Inthenextsubsection,weintroduceanoveldatastructure,calledreactionfunctions, that succinctly characterizes the local information of a robot for executing a complex taskatanygivenexecutiontime. InSubsection5.3.3,wethenintroduceanauctionalgo- rithm for solving complex-task allocation problems with disjoint coalitions, in which robots construct and submit reaction functions on unallocated tasks to the auctioneer and the auctioneer allocates one additional (simple or complex) task to robots in each roundsothatthecurrentteamcostincreasestheleast. Beforeweproceedtodescribethe concept of reaction functions, we describe our assumptions on complex-task allocation problemswithdisjointcoalitions: • AssumptionA: Theauctioneerhastoevaluateallpossiblequalifiedcoalitionsof eligible robots for allocating a complex task. The number of coalitions is expo- nentialinthecoalitionsizeofthecomplextask. Thus,weassumethatthelargest coalitionsizeofalltasksissmall,namely,boundedbyaconstantD. • AssumptionB: Weassumethateachrobotcanbeallocatedatmostonecomplex task. Forexample,afireenginemightonlyhaveenoughwatertohelpextinguish onebigfiretogetherwithotherfireengines. However,itmightstillbeabletohelp rescuesurvivorsandthusexecuteseveralsimpletasks. 137 5.3.2 ReactionFunctions Now we introduce the concept of reaction functions, explain how to construct and use them, and finally describe how to approximate them if necessary. In this subsection, we study how to determine the optimal execution time t of a given complex task x for a given qualified coalition P of d(x) eligible robots so that the resulting team cost after allocating taskx to coalitionP with execution timet increases the least. Assume that each robot r∈ P has been allocated a set of simple tasks X r s (without execution times) and has not yet been allocated any complex task (since it would not be eligible forexecutingcomplextaskxotherwise). ANaiveApproach Since robots in the given coalition P have to agree on the execution time of complex task x, they cannot locally determine the optimal execution time of the complex task andthushavetoprepareforallpossibilities. Insteadofbiddingasinglenumericalvalue on the complex task as in SK auctions, a robot now bids its robot costs of executing complextaskxatallpossibleexecutiontimes. Weproposethefollowingprocedurefor robotsandtheauctioneertodeterminetheoptimalexecutiontimeofcomplextaskxfor thecoalitionP ofrobots. • Bid-Construction Phase: Each robot r∈ P determines its robot costF r x (t) := c robot r (X r s ,{x},{x←t})forexecutingcomplextaskxattimetfromnowandall simple tasks in X r s at their optimal execution times.F r x (t) is a reaction function that maps any execution time 0≤ t <∞ to the robot costc robot r (X r s ,{x},{x← t}). Robotr thenbidsthereactionfunctionF r x (t)ontaskxtotheauctioneer. 138 • Winner-Determination Phase: For each reaction functionF r x (t) submitted by robotroncomplextaskx,theauctioneerfirstconstructsthefollowingevaluation functionbasedontheteamobjective: V r x (t) :=    F r x (t)−c robot r (X r s ,∅,∅) (MiniSumandMiniLat) F r x (t) (MiniMax) andthendeterminestheoptimalexecutiontimetofcomplextaskxasfollows: t = arg min 0≤t<∞ c team r∈P V r x (t). Theorem5.3.1 Allocating complex taskx to all robotsr∈ P with execution timet = argmin 0≤t<∞ c team r∈P V r x (t) (as determined by the auctioneer) results in the smallest team costofallpossibleexecutiontimesofthecomplextask. Proof: FortheMiniSumandMiniLatteamobjectives,c team r∈P V r x (t) = ∑ r∈P (F r x (t)− c robot r (X r s ,∅,∅))istheincreaseinthesumoftherobotcostsoftherobotsincoalitionP. Thus,itisalsotheincreaseinthesumoftherobotcostsofallrobotsinRsincenoother robot is allocated a new task. Let c be the sum of the robot costs of all robots before the allocation. Then,c team r∈P V r x (t)+c is the sum of the robot costs of all robots after the allocation. Minimizingc team r∈P V r x (t)thusalsominimizesthesumoftherobotcostsofall robots(theteamcostfortheMiniSumandMiniLatteamobjectives)aftertheallocation. For the MiniMax team objective, c team r∈P V r x (t) = max r∈P F r x (t) is the largest robot cost of the robots in coalition P after the allocation. Let c be the largest robot cost of any robot before the allocation. Since the allocation of a task to a robot cannot decrease its robot cost, max(c team r∈P V r x (t),c) is the largest robot cost of any robot after the allocation. Minimizingc team r∈P V r x (t)thusalsominimizesthelargestrobotcostofanyrobot(theteam costfortheMiniMaxteamobjective)aftertheallocation. 139 Theorem 5.3.1 proved that the execution time t determined by the auctioneer is indeedtheoptimalexecutiontimefortherobotsincoalitionP toexecutecomplextask x. However,therearetwomaintechnicaldifficultieswithapplyingthisapproach: • Difficulty 1: In order to construct reaction functionF r x (t), robot r has to deter- mine its robot costc robot r (X r s ,{x},{x← t}) for all execution times 0≤ t <∞. Therobothastosolveaninfinitenumberofoptimizationproblems,andtheresult- ingreactionfunctionconsistsofaninfinitenumberofnumericalvalues. • Difficulty 2: In order to determine the optimal execution time of the complex task,theauctioneerhastoevaluateaninfinitenumberofexecutiontimes. Inthefollowing,weaddressthesedifficultiestomaketheconstructionandusageof reactionfunctionsefficient. ConstructingReactionFunctions We first study how a robot r∈ P can construct its reaction functionF r x (t) for com- plextaskxwithoutsolvinganinfinitenumberofoptimizationproblems. Oneimportant insight is that there are only a finite number of execution orders of robot r to execute its allocated simple tasks and complex task x, although there are an infinite number of possible execution times of task x. The key idea of our approach is that we construct reactionfunctionF r x (t)basedontheseexecutionordersoftherobot. Beforewedescribe theprocedureforconstructingreactionfunction, wewanttoremindreadersofthecon- cept of reaction functions and the definition of robot cost: The reaction functionF r x (t) istherobotcostc robot r (X r s ,{x},{x← t})ofrobotr forexecutingitscomplextaskxat any given execution time t and its simple tasks in X r s at their optimal execution times. The robot cost c robot r (X r s ,{x},{x← t}) is the smallest task-completion time (for the MiniSum and MiniMax team objectives) or cumulative latency (for the MiniMax team 140 r :Robots p: VisitOrders t p :VisitTime c p : RobotCost n p s :Numberof ofx 5 Task-CompletionTime CumulativeLatency SimpleTargets (MiniSumandMiniMax) (MiniLat) VisitedAfterx 5 r 1 x 5 x 1 x 2 2 13 21 2 r 1 x 5 x 2 x 1 2 13 21 2 r 1 r 1 x 1 x 5 x 2 8 13 20 1 r 1 x 2 x 5 x 1 8 13 20 1 r 1 x 1 x 2 x 5 14 14 19 0 r 1 x 2 x 1 x 5 14 14 19 0 r 2 x 5 x 3 x 4 3 12 21.5 2 r 2 x 5 x 4 x 3 3 12 21.5 2 r 2 r 2 x 3 x 5 x 4 7 12 17.5 1 r 2 x 4 x 5 x 3 7 12 17.5 1 r 2 x 3 x 4 x 5 11 11 13.5 0 r 2 x 4 x 3 x 5 11 11 13.5 0 Table5.2: VisitOrdersofRobotsforExampleinFigure5.1 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Themeta-functionforvisitorders r 1 x 5 x 1 x 2 andr 1 x 5 x 2 x 1 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Themeta-functionforvisitorders r 1 x 1 x 5 x 2 andr 1 x 2 x 5 x 1 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Themeta-functionforvisitorders r 1 x 1 x 2 x 5 andr 1 x 2 x 1 x 5 Figure5.3: Meta-FunctionsofRobotr 1 forComplexTargetx 5 (MiniMaxTeamObjec- tive) objective) of robot r so that robot r executes its complex task x at the given execution timetanditssimpletasksinX r s attheiroptimalexecutiontimes. Robotr constructsreactionfunctionF r x (t)asfollows: 1. Robotr constructs all possible execution orders of its allocated simple tasks and complex task x. LetP r be the set of these execution orders. Consider again the multi-robot routing example as shown in Figure 5.1. Assume that robot r 1 is currently allocated simple targets X r 1 s = {x 1 ,x 2 } and robot r 2 is currently allocated simple targetsX r 2 s ={x 3 ,x 4 }. Table 5.2 tabulates all six possible visit ordersofeachrobotforvisitingitsallocatedsimpletargetsandcomplextargetx 5 . 141 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Themeta-functionforvisitorders r 2 x 5 x 3 x 4 andr 2 x 5 x 4 x 3 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Themeta-functionforvisitorders r 2 x 3 x 5 x 4 andr 2 x 4 x 5 x 3 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Themeta-functionforvisitorders r 2 x 3 x 4 x 5 andr 2 x 4 x 3 x 5 Figure5.4: Meta-FunctionsofRobotr 2 forComplexTargetx 5 (MiniMaxTeamObjec- tive) 2. For each execution order p∈P r , robot r finds the execution time t p of com- plex task x and calculates the resulting robot cost c p under the assumption that it executes all tasks in execution order p without waiting for any task. Robot r also determines the number of simple tasks n p s executed after complex task x in execution order p for the MiniLat team objective. Table 5.2 tabulates the visit timeofcomplextargetx 5 andthecorrespondingrobotcostforeachvisitorderfor themulti-robotroutingexampleshowninFigure5.1. Table5.2alsotabulatesthe numberofsimpletargetsvisitedaftercomplextargetx 5 foreachvisitorder. 3. Robot r constructs a meta-functionF r,p x (t) for each execution order p∈P r as follows: F r,p x (t) :=    c p +t−t p ift≥t p (MiniSumandMiniMax) c p +(n p s + 1 d(x) )(t−t p ) ift≥t p (MiniLat). For the sake of convenience, we defineF r,p x (t) :=∞ if t < t p . Meta-function F r,p x (t) characterizes the robot cost of robot r for executing complex task x at time t if robot r has to execute its tasks in execution order p. As defined in the previous step, t p is the execution time of complex task x if robot r executes its 142 tasks in execution order p without waiting for any task. For any execution time t < t p , robot r cannot execute complex task x at time t if it has to execute its tasksinexecutionorderp. Nowletusconsiderthevalueofthismeta-functionfor any execution timet≥ t p . For the MiniSum and MiniMax team objectives,c p is thetask-completiontimeofrobotr ifrobotr executesitstasksinexecutionorder p without waiting for any task. In order to execute complex task x at the given execution time t, robot r can wait t−t p time units for complex task x and then executeitandtheremainingsimpletasksinexecutionorderp. Sincerobotr does not need to wait for any simple task, its task-completion time increases byt−t p time units. Thus, the resulting robot cost is c p +t−t p for robot r if it executes complex task x at time t≥ t p and executes its tasks in execution order p. For the MiniLat team objective, c p is the cumulative latency of robot r (that is, the sumofthenormalizedlatenciesoftasks)ifrobotr executesitstasksinexecution order p without waiting for any task. In order to execute complex task x at the given execution timet, robotr can waitt−t p time units for complex taskx and then execute it and the remaining simple tasks in orderp. The execution times of complextaskxandallsimpletasksafteritinexecutionorderpincreasebyt−t p time units. Remember that the normalized latency of a task is its execution time divided by its coalition size, and the coalition size of a simple task is one. Thus, theresultingrobotcostisc p +n p s (t−t p )+ 1 d(x) (t−t p ) =c p +(n p s + 1 d(x) )(t−t p ) for robot r if it executes complex task x at time t≥ t p and executes its tasks in executionorderp. Consideragainthemulti-robotroutingexampleshowninFigure5.1fortheMini- Maxteamobjective. Figures5.3and5.4showthemeta-functionsofrobotsr 1 and r 2 respectivelyforcomplextargetx 5 forthevisitorderstabulatedinTable5.2. 143 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Thereactionfunctionofrobotr 1 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Thereactionfunctionofrobotr 2 Figure 5.5: Reaction Functions of Robots for Complex Target x 5 (MiniMax Team Objective) 4. Robotr determinesthevalueofitsreactionfunctionF r x (t)astheminimumofall meta-functionsF r,p x (t) constructed in the previous step since each meta-function characterizesitsrobotcostifitexecutesitstasksinoneparticularorder: F r x (t) = min p∈P r F r,p x (t). Consider again the multi-robot routing example in Figure 5.1 for the MiniMax teamobjective. Figure5.5showsthereactionfunctionsofrobotsr 1 andr 2 respec- tively for complex target x 5 , resulting from minimizing the values of the meta- functionsshowninFigures5.3and5.4. Thecorrectnessofourapproachisgivenbythefollowingtheorem: Theorem5.3.2 The robot cost of robot r for executing complex task x at time t is F r x (t) = min p∈P rF r,p x (t). Proof: Consider any execution time t, and let c be the robot cost of robot r for executingcomplextaskxattimetanditssimpletasksatoptimaltimes. Ifc =∞,then there does not exist any execution order of tasks in which robotr can execute complex task x at time t or, equivalently, t < t p for each execution order p∈P r . Thus, c = 144 ∞ = min p∈P rF r,p x (t). If c̸=∞, we first prove min p∈P rF r,p x (t)≥ c by contradiction asfollows: Ifmin p∈P rF r,p x (t)<c,thenthereexistsarobotcostofrobotrforexecuting its tasks in some execution order and complex taskx at timet that is less thanc, which contradicts the definition of the robot cost c. Now we prove min p∈P rF r,p x (t)≤ c as follows: We use p ∗ to denote an execution order of robot r with robot cost c if robot r executes tasks in this order and complex task x at time t (such an execution order exists since c̸=∞). We assume that robot r does not wait for any simple task. (If it does, then we can either move the wait times of robot r for the simple tasks that come in execution order p ∗ before the complex task to the complex task or reduce the robot costbyremovingtheunnecessarywaittimesofrobotrforthesimpletasksthatcomein execution orderp ∗ after the complex task.) We usew ∗ to denote the wait time of robot r at complex task x and n ∗ to denote the number of simple tasks after complex task x in execution order p ∗ . Consider the meta-functionF r,p ∗ x (t) of robot r for execution order p ∗ . Since robot r waits only for complex task x in execution order p ∗ , we have w ∗ =t−t p ∗ andc =c p ∗ +w ∗ =c p ∗ +t−t p ∗ =F r,p ∗ x (t)fortheMiniSumandMiniMax team objectivesandc = c p ∗ +(n p ∗ s + 1 d(x) )w ∗ = c p ∗ +(n p ∗ s + 1 d(x) )(t−t p ∗ ) =F r,p ∗ x (t) for the MiniLat team objective. Given that the setP r contains all possible execution orders of robot r, we have p ∗ ∈P r and thus min p∈P rF r,p x (t)≤F r,p ∗ x (t) = c. Thus, min p∈P rF r,p x (t) =c. Each meta-functionF r,p x (t) of robot r is a linear segment as shown in Figures 5.3 and 5.4. Thus, the reaction functionF r x (t) of robot r consists of at most|P r | linear segments. Sincethenumberofsimpletasks|X r s |isfinite,thereareonlyafinitenumber ofpossibleexecutionordersinP r . Asaresult,robotsneedtosolveonlyafinitenumber ofoptimizationproblemstoconstructtheirreactionfunctions,andtheresultingreaction functionscontainonlyafinitenumberofsegments. 145 DeterminingtheOptimalExecutionTime Nowwe address the second difficultyof our naive approach, namely, how to determine theoptimalexecutiontimeofcomplextaskxforagivencoalitionP ofrobots. Giventhe reaction functionF r x (t) and the corresponding evaluation functionV r x (t) of each robot r∈ P, the auctioneer has to evaluate all possible execution times of complex taskx to determinetheonethatminimizestheteamcost: t = arg min 0≤t<∞ c team r∈P V r x (t). Theminimizationisover 0≤ t <∞. Thefollowingtheorem,however,showsthatthis isunnecessary. Theorem5.3.3 For each robot r∈ P and each execution order p∈P r , let t p be the executiontimeofcomplextaskxifrobotrexecutesitstasksinexecutionorderpwithout waitingforanytask. LetT bethesetofalltheseexecutiontimest p . Then, min 0≤t<∞ c team r∈P V r x (t) = min t∈T c team r∈P V r x (t). Proof: Let t̸∈ T be the execution time of complex task x that minimizes the left- hand side, which is then strictly smaller than the right hand side (and thus not infinity) sinceitminimizesoveralargersetofexecutiontimes. c team r∈P V r x (t)cannotincreasewhen decreasing t by an infinitesimal amount since none of the functionsV r x (t) can increase unlesst∈T. Thus,onecandecreasetuntilt∈T∪{0}withoutincreasingc team r∈P V r x (t). However,theteamcostatt = 0 isinfinityunless 0∈ T. Thus, onecandecreasetuntil t∈T withoutincreasingc team r∈P V r x (t). Consider again the multi-robot routing example shown in Figure 5.1 for the Mini- Maxteam objective. Giventhe reaction functionsof robotsshownin Figure5.5, which 146 arealsotheirevaluationfunctions,theauctioneerneedstominimizeonlyoverthesetof visittimesT ={2,3,7,8,11,14}ofcomplextargetx 5 insteadofallpossiblevisittimes. ThesetsP r containsonlyafinitenumberofexecutionordersandthereareonlyafinite number of robots in a coalition. Thus, setT contains only a finite number of execution times of complex taskx that the auctioneer needs to evaluate in order to determine the optimalexecutiontime. ApproximatingReactionFunctions Although the reaction function of robot r for complex task x contains only a finite numberofsegments,thenumberofsegmentsinitsreactionfunctionisstillexponential in the number of simple tasks in X r s because the number of possible execution orders of tasks is exponential in the number of simple tasks in X r s . Similarly, the number of possibleexecutiontimesofcomplextaskxthattheauctioneerneedstoevaluateislinear in the number of robots but exponential in the largest number of simple tasks allocated to any robot in coalition P. Thus, the computation and communication of reaction functionsisstilltime-intensiveifthecardinalityofX r s isnotsmallforanyrobotr∈P. Nowweshowhowtoapproximateareactionfunctionsothatitcontainsonlyaconstant numberofsegments: 1. Robot r finds a sufficiently large execution time e so that it can execute all allo- cated tasks in X r s ∪{x} in any execution order (without waiting for any task) beforetimee. Forexample, ∑ x ′ ∈X r s ∪{x} (d(r,x ′ )+d(x ′ ,r))+ϵisasimplechoice ofe in a multi-robot routing problem, whered(r,x ′ ) is the travel time of robotr from its initial location to any targetx ′ ∈ X r s ∪{x},d(x ′ ,r) is the travel time of robotrfromtargetx ′ backtoitsinitiallocation,andϵisasmallpositiveconstant. Robot r then divides the time interval [0,e) evenly intoG time intervals [s j ,e j ) foragivendiscretizationgranularityG. 147 s := (x (1) ...x (l) )(randomly); 1 (c,t,p) :=Evaluate(s); 2 forq :=⌊l/2⌋...0do 3 forr := 1...l−q do 4 s ′ :=s; 5 remove(x (r) ...x (r+q) )froms ′ ; 6 foreachpositionsins ′ do 7 s ′′ :=insert(x (r) ...x (r+q) )atpositionsintos ′ ; 8 (c ′ ,t ′ ,p) :=Evaluate(s ′′ ); 9 ifc ′ <cthen 10 s :=s ′′ ;c :=c ′ ; 11 breakloopsandgotoline4; 12 end 13 s ′′ :=insert(x (r+q) ...x (r) )atpositionsintos ′ ; 14 (c ′ ,t ′ ,p) :=Evaluate(s ′′ ); 15 ifc ′ <cthen 16 s :=s ′′ ;c :=c ′ ; 17 breakloopsandgotoline4; 18 end 19 end 20 end 21 end 22 (c,t,p) :=Evaluate(s); 23 returnp; 24 Explanation: This algorithm implements the Or-opt heuristic [Or76] to approximately determine the execution orderp of robotr toexecuteitstasksinX r s ∪{x}withthesmallestrobotcostcsuchthat: i)theexecutiontimetofcomplextaskxisintimeinterval [s j ,e j ) and ii) robotr does not wait for any task. Lets be a random permutation of the simple tasksX r s = {x 1 ...x l } that are allocatedtorobotr. FunctionEvaluate(s)insertsthecomplextaskxatallpositionsintosandcalculatestheresultingrobotcost of robotr to execute all tasks in the given execution order without waiting for any task. It then returns the triple of the robot cost c,theexecutiontimetofthecomplextaskx,andtheexecutionorderp,minimizedoverallcases(ifany)wheretheexecutiontime of complex task x falls into time interval [s j ,e j ). If there are no such cases, it returns infinity for the robot cost, infinity for the executiontimeofthecomplextaskx,andNULLfortheexecutionorder. TheOr-optheuristictakesallsubsequencesoflengthsone to⌊l/2⌋+1 ofs and inserts them in both their original and reserved order into all positions in the remaining sequence ofs until therobotcostreturnedbyEvaluate(s ′′ )fortheresultingpermutations ′′ issmallerthantherobotcostreturnedbyEvaluate(s). Itthensetsstotheresultingpermutationandrepeatstheprocess. Figure5.6: Or-OptApproximationAlgorithm 2. Robotr solvesthefollowingoptimizationproblemforeachtimeinterval [s j ,e j ), namely,todeterminetheexecutionorderp j ofitstasksX r s ∪{x}withthesmallest robotcostsuchthat: 1)robotrexecutescomplextaskxatsometimet p j ∈ [s j ,e j ) and 2) it does not wait for any task. (Such an execution order does not always exist.) This optimization problem is a special case of job scheduling with time windows [CL87, LRS04]. In our experiments, we use a version of the Or-opt 148 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Theapproximatereactionfunctionofrobotr 1 withe = 14andG =2 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 5 robot cost (= task-completion time) Theapproximatereactionfunctionofrobotr 2 withe = 11andG =2 Figure 5.7: Approximate Reaction Functions of Robots for Complex Target x 5 (Mini- MaxTeamObjective) heuristic [Or76] to solve it approximately, as described in Figure 5.6. Let the set P r bethesetofexecutionordersp j foundforalltimeintervals [s j ,e j ). 3. The remaining three steps of robot r of constructing the approximate reaction functionareidenticaltoSteps2-4oftheprocedureforconstructingreactionfunc- tionsifonereplacesP r with P r everywhere. Clearly, P r ⊆P r , and P r =P r if robot r uses an infinite number of time inter- vals and solves the optimization problems optimally in Step 2. In general, approximate reactionfunctionsneednotbeidenticaltotheidealreactionfunctions. Forexample,Fig- ure 5.7 shows the approximate reaction functions of robots r 1 and r 2 for discretization granularityG = 2,wheretherobotssolvetheoptimizationproblemsinStep2optimally. Theseapproximatereactionfunctionsaredifferentfromthereactionfunctionsshownin Figure5.5. Since| P r |≤G,anapproximatereactionfunctioncontainsatmostG andthusacon- stantnumberofsegments 3 ,andthetotalnumberofexecutiontimesthattheauctioneer 3 ThevalueofthediscretizationgranularityG canbearbitrarilydeterminedbytheusers,forexample, itcanbelinearinthenumberofsimpletasks. Forthesakeofsimplicity,wechooseG tobeaconstantin thisdissertation. 149 needs to evaluate is linear in the number of robots in the coalition and independent of thenumberofsimpletasksallocatedtorobots. 5.3.3 SequentialAuctionswithReactionFunctions In the previous subsection, we proposed reaction functions to characterize the local informationofrobotsforexecutingagivencomplextaskatanygiventime,anddemon- strated how the auctioneer determines the optimal execution time of a given complex taskforagivencoalitionofrobots. Now, wedevelopanewauctionalgorithmforsolv- ingcomplex-taskallocationproblemswithdisjointcoalitions,calledsequentialauctions withreactionfunctions,inwhicheachrobotbidsreactionfunctionsonunallocatedtasks and the auctioneer allocates one additional task to robots so that the current team cost increasestheleastineachround. A sequential auction with reaction functions consists of multiple rounds to allocate (simple and complex) tasks to robots. Initially, all tasks are unallocated. During each round,allrobotsbidoneachunallocatedtaskareactionfunctionandthentheauctioneer allocates one additional task to robots so that the current team cost increases the least. Consider any round of the sequential auction with reaction functions. Assume that the current allocation of each robot r∈ R is (X r s ,X r c ,C r ), where X r s is the set of simple tasks andX r c is the set of complex tasks that have been allocated to robotr in previous rounds. Let X r = X r s ∪X r c . Thus, U = X\ ∪ r∈R X r is the set of unallocated tasks. We now explain exactly which bids robots submit and how the auctioneer makes the allocationdecision. • Bid-ConstructionPhase: Robotr∈ R considers each taskx∈ U: Letx c be its allocatedcomplextaskifwetentativelyallocatetaskxtorobotr. (x c iswrittenas NULL if such complex task does not exist). If robotr is eligible for taskx, then itconstructsareactionfunctionF r,x x c (t)asfollows: 150 –F r,x x c (t) =c robot r (X r s ,{x},{x←t})iftaskxiscomplex; –F r,x x c (t) = c robot r (X r s ∪{x},{x c },{x c ← t}) if task x is simple and x c ̸= NULL; –F r,x x c (t) =c robot r (X r s ∪{x},∅,∅)iftaskxissimpleandx c = NULL. 4 andbidsthisreactionfunctionF r,x x c (t)ontaskx. • Winner-Determination Phase: The auctioneer keeps the following informa- tion for each robot r ∈ R: 1) its current allocation (X r s ,X r c ,C r ), 2) its current robot cost c robot r (X r s ,X r c ,C r ), and 3) its reaction functionF r x r(t) = c robot r (X r s ,{x r },{x r ← t}) if robot r has been allocated a complex task x r in previousrounds. We first introduce two important concepts that the auctioneer has to use when it evaluatestheallocationofataskx∈U toacoalitionP ofrobots: – The related complex task of the allocation of task x to coalition P is the complextaskallocatedtoanyrobotinP iftheauctioneertentativelymakes this allocation, which is written as NULL if such a complex task does not exist. – TherelatedrobotsoftheallocationoftaskxtocoalitionP aretherobotsthat areallocatedtoexecutetherelatedcomplextaskiftheauctioneertentatively makesthisallocation,whichisdefinedtobeP iftherelatedcomplextaskis NULL. 4 AlthoughF r;x NULL (t) = c robot r (X r s ∪{x};∅;∅) is a numerical value, we still write it as a (trivial) constantfunctionofexecutiontimesforthesakeofconvenience. 151 Nowweshowhowtheauctioneerallocatesoneadditionaltaskx∈U torobotsso that the current team cost increases the least. Remember that solving a complex- task allocation problem involves solving two subproblems: 1) which coalition of robotsshouldexecuteagiventask;and2)whenagivencoalitionofrobotsshould executeagiventask. – Solutionofthefirstsubproblem: For each unallocated simple or complex task x∈ U, the auctioneer first constructs the set P(d(x)) that contains all qualifiedcoalitionsofd(x)differenteligiblerobotsforit. Foreachcoalition P∈P(d(x)),theauctioneerthendeterminestheevaluationcostc eval x,P ofthe allocation of task x to coalition P and the optimal execution time t x,P for therelatedcomplextaskofthisallocation(iftherelatedcomplextaskisnot NULL) as described below in the solution of the second subproblem. After that,theauctioneerdetermines (x,P) := argmin x∈U,P∈P(d(x)) c eval x,P . Finally, the auctioneer allocates the task x to all robots in P and sets the execution time of the related complex task (if any) tot x,P for all related robots of this allocation. – Solutionofthesecondsubproblem: Letx ′ betherelatedcomplextaskand R ′ bethesetofrelatedrobotsfortheallocationoftaskxtocoalitionP. The auctioneer first constructs the following evaluation function for each robot r∈R ′ : V r x ′(t) :=    F r x ′(t)−c robot r (X r s ,X r c ,C r ) (MiniSumandMiniLat) F r x ′(t) (MiniMax). 152 ThereactionfunctionF r x ′(t)ofrobotr isavailabletotheauctioneer: F r x ′(t) :=    F r,x x c (t) ifr∈P F r x r(t) ifr / ∈P, whereF r,x x c (t) is the reaction function submitted by robot r on task x in the bid-construction phase andF r x r(t) is the reaction function of robotr for its currently allocated complex task x r . Finally, the auctioneer determines c eval x,P = min 0≤t<∞ c team r∈R ′V r x ′(t) as the evaluation cost of this allocation and t x,P = argmin 0≤t<∞ c team r∈R ′V r x ′(t) as the execution time of the related com- plextaskx ′ ofthisallocationifx ′ ̸= NULL. The following theorem shows that sequential auctions with reaction functions achievetheperformanceprincipleofcomplex-taskauctions. Theorem5.3.4 Allocating task x to all d(x) robots r∈ P with execution time t x,P of the related complex task (if any) as determined by the auctioneer results in the smallest teamcostamongallpossibleallocationsofanyunallocatedtaskx ′′ ∈U toanyqualified coalitionofd(x ′′ )differenteligiblerobots. Proof: Letx ′ be the related complex task andR ′ be the set of related robots for an allocation of taskx∈ U to coalitionP∈ P(d(x)). We first show that the robot cost of any robotr∈ R\R ′ is not changed by this allocation since: i) this allocation does not allocate any new task to robot r given that r / ∈ P and P ⊂ R ′ ; and ii) this allocation does not change the commitment (if any) of robotr given that only the execution time oftherelatedcomplextaskx ′ maybechangedafterthisallocation. Consider the MiniSum and MiniLat team objectives. The evaluation cost c eval x,P of the allocation of task x to coalition P is min 0≤t<∞ ∑ r∈R ′ V r x ′(t) = min 0≤t<∞ ∑ r∈R ′ (F r x ′(t)− c robot r (X r s ,X r c ,C r )), which is the minimal increase in the 153 sum of the robot costs of all related robots in R ′ . Thus, it is also the minimal increase in the sum of the robot costs of all robots inR since the robot costs of robots inR\R ′ arenotchanged. Asaresult,theminimizationmin x∈U,P∈P(d(x)) c eval x,P alsominimizesthe increase of the team cost among all possible allocations. Now consider the MiniMax team objective. The evaluation cost c eval x,P of the allocation of task x to coalition P is min 0≤t<∞ max r∈R ′V r x ′(t) = min 0≤t<∞ max r∈R ′F r x ′(t), which is the largest robot cost ofanyrelatedrobotinR ′ aftertheallocation. Letcbethelargestrobotcostofanyrobot before the allocation. max(c eval x,P ,c) is the largest robot cost of any robot after the allo- cationsincetherobotcostsofrobotsinR\R ′ arenotchangedandtheteamcostnever decreases after making a new allocation. Thus, the minimization min x∈U,P∈P(d(x)) c eval x,P alsominimizes max(c eval x,P ,c)andthustheteamcostamongallpossibleallocations. 5.4 OverlappingCoalitions In the previous section, we introduced the concept of reaction functions that each char- acterizes the robot cost of a robot for executing a single complex task at any given exe- cution time and showed how the auctioneer can determine the optimal execution time of a single complex task based on reaction functions. In this section, we study how to generalize the concept and the usage of reaction functions if a robot can be allocated multiplecomplextasks. 5.4.1 Overview Assume that each robot r∈ R has been allocated a set of simple tasks X r s ⊆ X and a set of complex tasks X r c ⊆ X. We consider how to determine the optimal execu- tion times of all allocated complex tasks in ∪ r∈R X r c so that the resulting team cost is smallest. Consider any robot r∈ R, and let x r = (x r 1 ,...,x r nr ) be the vector of its 154 allocated complex tasks X r c (in an arbitrary order), t r = (t r 1 ,...,t r nr ) be the vector of thecorrespondingexecutiontimesofthesecomplextasks,andx r ←t r bethesetofits commitments{x r 1 ← t r 1 ,...,x r nr ← t r nr }. Then, the robot cost of robotr for executing complextasksx r atexecutiontimest r isc robot r (X r s ,X r c ,x r ←t r ). Letx = (x 1 ,...,x n ) be the vector of all allocated complex tasks in ∪ r∈R X r c and t = (t 1 ,...,t n ) be the vector of the corresponding execution times of these complex tasks. We define a pro- jection functionf r (v) to project any vectorv of complex tasks onto the complex tasks allocated to robot r∈ R in the same order. For example, x r = f r (x) andt r = f r (t). Thus,theoptimalexecutiontimesofallcomplextasksxcanbedeterminedas: arg min t∈R n + c team r∈R c robot r (X r s ,X r c ,x r ←f r (t)), whereR + isthesetofallnon-negativerealnumbers. Before we generalize the concept of reaction functions, we first have to revisit the concept of related robots introduced in the previous section. In case of disjoint coali- tions, where each robot can execute at most one complex task, a bundle of robots are related if they are all allocated to execute a common complex task. The definition of relatedrobotsismorecomplicatedincaseofoverlappingcoalitions: Forthegivenallo- cations of complex tasks X r c to robots r∈ R, we construct a bipartite graph of robots R andcomplextasks ∪ r∈R X r c ,whereanedgeconnectsarobotr andacomplextaskx ifftaskxisallocatedtorobotr,thatis,x∈X r c . Tworobots(ortwocomplextasks)are related iff they are connected in the graph. This relationship partitions the set of robots R into a set of pairwise disjoint bundles B ={b 1 ,...,b l }, where each bundle b∈ B consistsoftherobotsthatarerelatedtoeachother 5 . Letx b = (x b 1 ,...,x b n b )bethevector of all complex tasks allocated to robots in bundleb andt b = (t b 1 ,...,t b n b ) be the vector 5 Bundlesareallowedtocontainsinglerobotsiftherearenocomplextasksallocatedtothoserobots. 155 ofthecorrespondingexecutiontimesoftheserelatedcomplextasks. Robotsindifferent bundles can determine the execution times of their allocated tasks independently since they don’t need to coordinate with others to determine these execution times. Thus, we have: min t∈R n + c team r∈R c robot r (X r s ,X r c ,x r ←f r (t r )) =c team b∈B ( min t b ∈R n b + c team r∈b c robot r (X r s ,X r c ,x r ←f r (t b ))). In other words, we can solve the following minimization problem independently for each bundle b∈ B to determine the optimal execution times of all allocated complex tasks: min t b ∈R n b + c team r∈b c robot r (X r s ,X r c ,x r ←f r (t b )). We also revise the assumptions made in case of disjoint coalitions to the case of overlappingcoalitionsasfollows: • AssumptionA: This assumption is identical to the one made in case of disjoint coalitions. We assume that the largest coalition size of all tasks is small, namely, boundedbyaconstantD. • AssumptionB ′ : We assume that the complex-task capacityq r c of each robotr∈ R is bounded by a small constantC, which extends the assumption made in case of disjoint coalitions where robots can execute at most one complex task. Thus, each robot needs to consider at mostC! different execution orders of its allocated complextasks. 156 x 4 r 1 x 1 r 2 r 3 x 2 x 3 Figure5.8: Example10ofMulti-RobotRouting 5.4.2 GeneralizedReactionFunctions Foreachrobotr∈RwithitsallocatedsimpletasksX r s anditsallocatedcomplextasks X r c , we generalize the concept of reaction functions to characterize its robot cost for executingthecomplextasksinX r c atanygivenexecutiontimesasfollows: F r X r c (t r ) :=c robot r (X r s ,X r c ,x r ←t r ). Letn r be the number of complex tasks inX r c . The domain of this generalized reaction functionF r X r c (t r ) isR nr + . It corresponds to all possible execution times of robot r for executing the n r complex tasks in X r c . There are an infinite number of optimization problems robot r needs to solve if it constructs the generalized reaction function by calculatingitsrobotcostforeverypossibleofvectorexecutiontimest r ∈R nr + . Similarly,theauctioneerneedstoevaluateallpossibleexecutiontimesoftherelated complex tasks that are allocated to bundle b to determine the optimal execution times. Letn b bethenumberofrelatedcomplextasksforbundleb. Then,theauctioneersolves thefollowingminimizationproblem: t b = arg min t b ∈R n b + c team r∈b F r X r c (f r (t b )). 157 In this subsection, we show how a robot r∈ b constructs its generalized reaction function by considering only a finite number of execution orders of its allocated tasks and how the auctioneer determines the optimal execution times by evaluating only a finite number of candidates. We also develop an approximation technique that makes the construction and usage of generalized reaction functions more efficient (when nec- essary). Figure 5.8 shows an example of multi-robot routing with complex targets in known terrain for the MiniMax team objective. It consists of three robots, three complex tar- gets and only one simple target. Robots need to form overlapping coalitions to visit the complex targets in this example. There still exists only one capability. Robots r 1 , r 2 and r 3 can provide one unit of this capability each. Targets x 1 , x 2 and x 3 require two units of this capability each, while target x 4 require only one unit. For all robots, the simple-target capacity is one and the complex-target capacity is two. Assume the current allocations of the robots are: robot r 1 has been allocated two complex targets X r 1 c ={x 1 ,x 3 } and no simple target; robot r 2 has been allocated two complex targets X r 2 c ={x 1 ,x 2 } and one simple target X r 2 s ={x 4 }; and robot r 3 has been allocated two complex targets X r 2 c ={x 2 ,x 3 } and no simple target. Obviously, all three robots arerelatedforthegivenallocations. Wewillusethisexampletoillustrateourapproach throughoutthissubsection. ConstructingGeneralizedReactionFunctions Nowweshowhowarobotr∈Rconstructsits(generalized)reactionfunctionF r X r c (t r ) for its allocated complex tasksX r c . In this subsection, we assume that robotr has been allocated at least one complex task, that is, X r c ̸=∅. (Otherwise, the reaction function F r ∅ (t r ) =c robot r (X r s ,∅,∅)isanumericalvalue.) 158 π: Permutations p: VisitOrders t p : Vectors c p : RobotCost n p s : Vectorsof ofComplex ofVisitTimes Task-CompletionTime CumulativeLatency Numbersof TargetsX r 2 c (MiniSumandMiniMax) (MiniLat) SimpleTargets r 2 x 1 x 2 x 4 (5,10) 13 20.5 (0,1) (x 1 ,x 2 ) r 2 x 1 x 4 x 2 (5,12) 12 17.5 (1,0) r 2 x 4 x 1 x 2 (11,16) 16 20.5 (0,0) r 2 x 2 x 1 x 4 (4,9) 13 19.5 (0,1) (x 2 ,x 1 ) r 2 x 2 x 4 x 1 (4,11) 11 14.5 (1,0) r 2 x 4 x 2 x 1 (10,15) 15 19.5 (0,0) Table5.3: VisitOrdersofRobotr 2 forExampleinFigure5.8 By definition, reaction functionF r X r c (t r ) characterizes the robot cost of robotr∈ b for executing its complex tasks in X r c at any possible execution times. It appears that robot r needs to solve an infinite number of optimization problems. However, there are onlyafinitenumberofexecutionordersforrobotr toexecuteitsallocated(simpleand complex) tasks inX r s ∪X r c . Robotr can construct its reaction function based on these execution orders, similar to the case of disjoint coalitions. Robotr performs the follow stepstoconstructitsreactionfunction: 1. Robotr constructsallpossibleexecutionordersofitsallocated(simpleandcom- plex)tasksinX r s ∪X r c . LetP r bethesetofalltheseexecutionorders. 2. Robot r constructs all possible permutations of its allocated complex tasks in X r c ={x r 1 ,...,x r nr }. Let r bethesetofallthesepermutations. Foreachpermu- tationπ = (x r π(1) ,...,x r π(nr) ) in r , letP r,π be the set of execution orders inP r inwhichthecomplextasksareexecutedintheordergivenbypermutationπ. Considerthemulti-robotroutingexampleshowninFigure5.8. Table5.3tabulates bothpermutationsofcomplextargetsX r 2 c ={x 1 ,x 2 }andallsixexecutionorders oftargetsx 1 ,x 2 andx 4 thatareallocatedtorobotr 2 . 3. For each permutation π ∈ r and each execution order p ∈ P r,π , robot r finds the vector of execution times t p = (t p π(1) ,...,t p π(nr) ) of the complex tasks π = (x r π(1) ,...,x r π(nr) ) and calculates the resulting robot costc p if it executes its 159 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 = t 2 t 1 t 2 (5, 12) (11, 16) (5, 10) (a) 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 (5, 10) ( , ) - ( , ) (b) 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 (5, 10) (c) Figure 5.9: Domains of Generalized Reaction Function of Robot r 2 (MiniMax Team Objective) allocated tasks in execution order p without waiting for any task. Robot r also findsthenumbern p s,i ofsimpletasksexactlyaftercomplextaskx r π(i) (andexactly before complex taskx r π(i+1) ifi < n r ) in execution orderp for the MiniLat team objective. Let n p s = (n p s,1 ,...,n p s,nr ) be the vector of these numbers of simple tasksbetweencomplextasksinexecutionorderp. For the multi-robot routing example shown in Figure 5.8, Table 5.3 tabulates the vector of visit times of complex targets and the robot cost of each visit order of robot r 2 . Table 5.3 also tabulates the vector of simple targets between complex targetsforeachvisitorder. 4. For each permutation π∈ r and each execution order p∈P r,π , robot r con- structsameta-functionF r,p π (t r π )whichcharacterizestherobotcostofrobotr ifit executesitsallocatedtasksinexecutionorderp,asfollows: • Function Domain: If robot r is required to execute its complex tasks in oneparticularexecutionorderπ = (x r π(1) ,...,x r π(nr) ),thenthevectorofthe execution times t r π = (t r π(1) ,...,t r π(nr) ) of the complex tasks has to satisfy t r π(nr) ≥ ...≥ t r π(1) ≥ 0. We define a boolean functionB(t r ) = true iff 160 t r nr ≥ ...≥ t r 1 ≥ 0 for any vectort r = (t r 1 ,...,t r nr ). Consider again robot r 2 in the multi-robot routing example shown in Figure 5.8. Figure 5.9(a) shows the domain (t 1 ≥ 0 and t 2 ≥ 0) of the reaction function of robot r 2 for its complex targetsx 1 andx 2 , and the shaded area (t 2 ≥ t 1 ≥ 0) shows allpossiblevectorsofvisittimes(t 1 ,t 2 )ofcomplextargetsx 1 andx 2 forthe permutation (x 1 ,x 2 ). Figure 5.9(a) also shows three vectors of visit times (5,10), (5,12) and (11,16) of robot r 2 for visiting complex targets x 1 and x 2 forthegivenpermutation (x 1 ,x 2 )astabulatedinTable5.3. The domain of the meta-function for execution order p∈P r,π is the set of all possible vectors t r π of execution times of the complex tasks if robot r executes its allocated tasks in execution orderp and waits only for complex tasks. For the sake of convenience, lett r π(0) = t p π(0) = 0. Given that robotr executesitstasksinexecutionorderpandwaitsonlyforcomplextasks,the executiontimet r π(i) ∈t r π ofcomplextaskx r π(i) ∈π mustbeatleastt r π(i−1) + t p π(i) −t p π(i−1) ,whichimpliest r π(i) −t r π(i−1) ≥t p π(i) −t p π(i−1) forall1≤i≤n r . (This constraint can be understood as follows: t r π(i) −t p π(i) is the total wait time of robot r for all complex tasks x r π(j) with 1≤ j≤ i, which must be non-decreasingini. Fromthis,wehavet r π(i) −t p π(i) ≥t r π(i−1) −t p π(i−1) ,which impliestheconstraint.) Thus,thedomainofthemeta-functionforexecution orderp∈P r,π isthesetofvectorsofexecutiontimest r π = (t r π(1) ,...,t r π(nr) ) that satisfy the domain constraintt r π(nr) −t p π(nr) ≥ ...≥ t r π(1) −t p π(1) ≥ 0, which can be written asB(t r π −t p ) = true. For example, Figure 5.9(b) shows a vector of visit times t r 2 (x 1 ,x 2 ) that satisfies this domain constraint of the meta-function of robot r 2 for the visit order p = (r 2 x 1 x 2 x 4 ). TheshadedareainFigure5.9(c)showsthewholedomainofthismeta- function. 161 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t p t 1 t 2 (5, 10) t 1 = 5 no wait at target x 1 (a) 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t p t 1 t 2 (5, 10) t 2 – t 1 = 5 no wait at target x 2 (b) Figure 5.10: Domain Boundaries of One Meta-Function of Robot r 2 (MiniMax Team Objective) • Function Domain Boundary: Now we study the domain boundaries of the meta-function for the execution order p∈P r,π . Since robot r is allo- cated n r complex tasks, the domain of each meta-function is part of the n r -dimensionalspaceR nr + andhasn r domainboundaries. Eachboundaryis characterizedbythefollowinghyperplaneinthen r -dimensionalspaceR nr + : t r π(nr) −t p π(nr) ≥ ...≥ t r π(i) −t p π(i) = t r π(i−1) −t p π(i−1) ≥ t r π(1) −t p π(1) ≥ t r π(0) −t p π(0) = 0foreach1≤i≤n r ,withthemeaningthatrobotr doesnot waitforcomplextaskx r π(i) sincetheconstraintt r π(i) −t p π(i) =t r π(i−1) −t p π(i−1) implies that the total wait time of robot r does not increase from the time when the robot executes complex task x r π(i−1) to the time when it executes complextaskx r π(i) . Forexample,Figure5.10showsthedomainboundariesofthemeta-function of robotr 2 for the visit orderp = (r 2 x 1 x 2 x 4 ) (whose domain is showninFigure5.9(c)). Therearetwoboundariesofthisdomain,andeach boundary is a straight line. The vector t p of visit times of targets (x 1 ,x 2 ) forthisvisitorderis (5,10). AsshowninFigure5.10(a),thefirstboundary restricts the visit time t 1 of complex target x 1 to be 5. Since the earliest 162 time for robot r 2 to arrive at complex target x 1 is 5 if robot r 2 visits its targets in visit orderp, robotr 2 does not wait at targetx 1 if its visit time of complextargetx 1 is5. Similarly,Figure5.10(b)showsthesecondboundary ofthedomainthatrestrictsthedifferencebetweenthevisittimesofcomplex targets x 1 and x 2 to be 5. Since the smallest travel time between targets x 1 andx 2 is5ifrobotr 2 visitsitstargetsinvisitorderp,robotr 2 doesnotwait at target x 2 if it visits complex target x 2 5 time units after it visits complex targetx 1 . • Function Value: The value of meta-functionF r,p π (t r π ) for execution order p∈P r,π is the robot cost of robot r for executing the complex tasks π at execution times t r π if robot r executes its tasks in execution order p. Let 1 d(π) bethevector ( 1 d(x r (1) ) ,..., 1 d(x r (nr) ) ) neededtocalculatethenormalized latenciesofcomplextasksfortheMiniLatteamobjective. Thus,wehave: F r,p π (t r π ) :=    c p +t r π(nr) −t p π(nr) iffB(t r π −t p ) =true (MiniSumandMiniMax) c p +(n p s + 1 d(π) )·(t r π −t p ) iffB(t r π −t p ) =true. (MiniLat) For the sake of convenience, we defineF r,p π (t r π ) :=∞ ifB(t r π − t p ) = false. For any vectort r π that satisfies the domain constraintB(t r π −t p ) = true, robot r can execute all tasks in execution order p and its complex tasks at execution times t r π without waiting for any simple tasks. Let us first consider the MiniSum and MiniMax team objectives: c p is the task- completiontimeofrobotrifitexecutesitsallocatedtasksinexecutionorder pwithoutwaitingforanytask. Thetotalwaittimeofrobotrforallcomplex tasksist r π(nr) −t p π(nr) . Sincerobotr doesnotwaitforsimpletasks,itstask- completion time increases by t r π(nr) −t p π(nr) . Thus, the resulting robot cost isc p +t r π(nr) −t p π(nr) . LetusnextconsidertheMiniLatteamobjective: c p is 163 thecumulativelatencyoftasksifrobotr executestheminexecutionorderp withoutwaitingforanytask. Theexecutiontimeofeachcomplextaskx r π(i) or any simple task between it and the next complex task x r π(i+1) (if i < n r ) in execution orderp increases byt r π(i) −t p π(i) . Thus, the cumulative latency of tasks increases by ∑ 1≤i≤nr (n p s,i + 1 d(x r (i) ) )· (t r π(i) −t p π(i) ), which can be writtenas (n p s + 1 d(π) )·(t r π −t p ). 5. For each permutation π∈ r , robot r constructs a permutation functionF r π (t r π ) whosedomainisthesetofvectorsofexecutiontimest r π forcomplextasksπ that satisfy the domain constraintB(t r π ) = true and whose value is determined as theminimumofthemeta-functionsF r,p π (t r π )forallexecutionordersp∈P r,π : F r π (t r π ) = min p∈P r; F r,p π (t r π ). 6. Finally, the generalized reaction functionF r X r c (t r ) of robot r is the collection of thepermutationfunctionsF r π (t r π )forallpermutationsπ∈ r . Thecorrectnessofourapproachisgivenbythefollowinglemmaandtheorem. Lemma5.4.1 For each permutation π ∈ r of the complex tasks allocated to robot r, the robot cost of robot r for executing its complex tasks π = (x r π(1) ,...,x r π(nr) ) at any given vector of execution times t r π = (t r π(1) ,...,t r π(nr) ) (withB(t r π ) = true) and its simple tasks at the optimal execution times is the value of the permutation function F r π (t r π ),namely, min p∈P r;F r,p π (t r π ). Proof: Letc ∗ betherobotcostofrobotr forexecutingcomplextasksπ atthegiven vectorofexecutiontimest r π . Ifc ∗ =∞,thentheredoesnotexistanyexecutionorderof tasksinwhichrobotrcanexecutethecomplextasksπatexecutiontimest r π forthegiven permutation π, that is,B(t r π −t p ) = false for each execution order p∈P r,π . Thus, 164 c ∗ =∞ = min p∈P r;F r,p π (t r π ). If c ∗ ̸=∞, then first prove min p∈P r;F r,p π (t r π )≥ c ∗ bycontradictionasfollows: If min p∈P r;F r,p π (t r π )<c ∗ ,thenthereexistsarobotcostof robotrforexecutingcomplextasksπattheexecutiontimest r π thatislessthanc ∗ ,which contradicts the definition of the robot cost c ∗ . Now we prove min p∈P r;F r,p π (t r π )≤ c ∗ as follows: Let p ∗ be an execution order of robot r that results in robot cost c ∗ . Let w r = (w r π(1) ,...,w r π(nr) ) be the vector of total wait times of robotr for complex tasks, where w r π(i) is the total wait time of robot r for all complex tasks x r π(j) with 1≤ j≤ i. Consider the meta-functionF r,p ∗ π (t r π ) of robot r for execution order p ∗ . We have t r π(nr) = t p ∗ π(nr) +w r π(nr) andF r,p ∗ π (t r π ) = c p ∗ +t r π(nr) −t p ∗ π(nr) = c p ∗ +w r π(nr) = c ∗ for the MiniSum and MiniMax team objectives. For the MiniLat team objective, we have t r π(i) =t p ∗ π(i) +w r π(i) foreach1≤i≤n r andF r,p ∗ π (t r π ) =c p ∗+(n p ∗ s + 1 d(π) )· (t r π −t p ∗ ) = c p ∗ + ∑ 1≤i≤nr (n p ∗ s,i + 1 d(x r (i) ) )·(t r π(i) −t p ∗ π(i) ) =c p ∗ + ∑ 1≤i≤nr (n p ∗ s,i + 1 d(x r (i) ) )·w r π(i) =c ∗ . Given thatp ∗ ∈P r,π , we have min p∈P r;F r,p π (t r π )≤F r,p ∗ π (t r π ) = c ∗ . Thus,F r,p π (t r π ) = c ∗ . Theorem5.4.1 The robot cost of robot r for executing its complex tasks in X r c = {x r 1 ,...,x r nr }atanygivenexecutiontimest r = (t r 1 ,...,t r nr )anditssimpletasksX r s at theoptimalexecutiontimescanbecalculatedfromitsreactionfunctionF r X r c (t r ). Proof: Sort the execution times t r = (t r 1 ,...,t r nr ) in increasing order to yield t r π = (t r π(1) ,...,t r π(nr) ) with B(t r π ) = true, which results in permutation π = (x r π(1) ,...,x r π(nr) ). Lemma 5.4.1 states that the robot cost of robot r for execut- ing complex tasks (x r π(1) ,...,x r π(nr) ) at execution times t r π = (t r π(1) ,...,t r π(nr) ) (with B(t r π ) = true) and its simplex tasks at the optimal execution times can be calculated as the value of the permutation functionF r π (t r π ), which is also the robot cost of robotr for executing complex tasks (x r 1 ,...,x r nr ) at execution timest r = (t r 1 ,...,t r nr ) and its simplextasksattheoptimalexecutiontimes. 165 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 3 (7, 13) (13, 7) Thereactionfunctionofrobotr 1 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 (5, 12) (11, 16) (5, 10) (9, 4) (11, 4) (15,10) Thereactionfunctionofrobotr 2 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 2 t 3 (4, 9) (10, 5) Thereactionfunctionofrobotr 3 Figure 5.11: Domains of (Generalized) Reaction Functions of Robots (MiniMax Team Objective) Forexample,Figure5.11showsthedomainsofthereactionfunctionsofrobotr 1 ,r 2 andr 3 forthe multi-robot routingproblem shownin Figure5.8. Each reaction function consists of two permutation functions. A permutation function contains multiple meta- functions if the robot has more than one visit order of targets for that permutation of complextargets(inotherwords,iftherobothasoneormoresimpletargetsallocated). DeterminingOptimalExecutionTimes Now we study how the auctioneer determines the optimal execution times of complex tasks. Given the (generalized) reaction functionsF r X r c (t r ) of each robotr in the bundle b of related robots, the optimal execution timest b = (t b 1 ,...,t b n b ) of the complex tasks x b = (x b 1 ,...,x b n b )inX r c thatareallocatedtobundlebcanbedeterminedas: t b = arg min t b ∈R n b + c team r∈b F r X r c (f r (t b )). The minimization is overR n b + , which corresponds to all possible execution times of n b complextasks. Thefollowingtheorem,however,showsthatthisisunnecessary. 166 ComplexTargets Permutations Allocations Scenarios r 1 doesnotwaitatx 1 ,r 2 doesnotwaitatx 2 ,andr 1 doesnotwaitatx 3 (x 1 ,x 2 ,x 3 ) X r 1 c ={x 1 ,x 3 } r 1 doesnotwaitatx 1 ,r 2 doesnotwaitatx 2 ,andr 3 doesnotwaitatx 3 (x 1 ,x 3 ,x 2 ) r 1 doesnotwaitatx 1 ,r 3 doesnotwaitatx 2 ,andr 1 doesnotwaitatx 3 {x 1 ,x 2 ,x 3 } (x 2 ,x 1 ,x 3 ) X r 2 c ={x 1 ,x 2 } r 1 doesnotwaitatx 1 ,r 3 doesnotwaitatx 2 ,andr 3 doesnotwaitatx 3 (x 2 ,x 3 ,x 1 ) r 2 doesnotwaitatx 1 ,r 2 doesnotwaitatx 2 ,andr 1 doesnotwaitatx 3 (x 3 ,x 1 ,x 2 ) X r 3 c ={x 2 ,x 3 } r 2 doesnotwaitatx 1 ,r 2 doesnotwaitatx 2 ,andr 3 doesnotwaitatx 3 (x 3 ,x 2 ,x 1 ) r 2 doesnotwaitatx 1 ,r 3 doesnotwaitatx 2 ,andr 1 doesnotwaitatx 3 r 2 doesnotwaitatx 1 ,r 3 doesnotwaitatx 2 ,andr 3 doesnotwaitatx 3 Table5.4: PermutationsandScenariosforExampleinFigure5.8 Theorem5.4.2 LetT be the set of vectors of execution timest b at which the robots in bundle b are able to execute the complex tasksx b without waiting for any simple tasks and for each complex task inx b there exists at least one robot in its coalition that does notwaitforit. Then, min t b ∈T c team r∈b F r X r c (f r (t b )) = min t b ∈R n b + c team r∈b F r X r c (f r (t b )). Proof: We first prove that the left-hand side cannot be smaller than the right-hand side since it minimizes over a larger set of vectors of execution times. Now we prove that the left-hand side cannot be larger than the right-hand side either: Let t b be the optimal execution times of robots for executing the complex tasks inx b . If there exists any simple task for which its allocated robot waits or any complex task for which all robots in its coalition wait, run the following procedure on this task x: First, find the smallest wait time s x > 0 of all robots for task x. Then, let each robot execute task x s x time units earlier. The resulting team cost does not increase since all robots execute their tasks no later than before. Repeat this procedure until robots do not wait for any simpletaskandforeachcomplextaskthereexistsatleastonerobotinitscoalitionthat does not wait for it. The resulting execution times belong to set T, and the resulting teamcostisnotlargerthanthatoftheoptimalexecutiontimes t b . NowweconsiderhowtoconstructthesetT. Rememberthateachdomainboundary of a meta-function in the reaction function of robotr∈ b specifies the execution times 167 atwhichrobotr doesnotwaitforoneofitsallocatedcomplextasks. Thus,eachvector ofexecutiontimesinsetT correspondstotheintersectionsofthedomainboundariesof themeta-functions. NowwedescribeexactlyhowtheauctioneerconstructssetT based onthereactionfunctionsF r X r c (t r )submittedbyeachrobotr∈b: 1. Theauctioneerconstructsallpossiblepermutationsπoftherelatedcomplextasks {x b 1 ,...,x b n b } that are allocated to (the robots in) bundle b. Consider again the multi-robotroutingexampleshowninFigure5.8. Giventhatrobotr 1 isallocated complextargets{x 1 ,x 3 },robotr 2 isallocatedcomplextargets{x 1 ,x 2 },androbot r 3 is allocated complex targets{x 2 ,x 3 }, all three complex targets x 1 ,x 2 and x 3 are related and there are six possible permutations of them, as tabulated in Table 5.4. 2. The auctioneer constructs all possible scenarios so that, for each complex task in x b ,thereisatleastonerobotinitscoalitionthatdoesnotwaitforit. Forexample, Table5.4tabulatesalleightpossiblescenariosforthemulti-robotroutingexample showninFigure5.8. 3. For each combination of a scenario and a permutationπ b = (x b π b (1) ,...,x b π b (n b ) ), the auctioneer generates a search tree for constructing execution times t b π b = (t b π b (1) ,...,t b π b (n b ) ) that belong to set T, starting with the root node (t b π b (1) ≥ 0,...,t b π b (n b ) ≥ t b π b (n b −1) ) at depth 0. The depth of the search tree is n b . Each node at any depth i with 1≤ i≤ n b has the form (t b π b (1) ,...,t b π b (i) ,t b π b (i+1) ≥ t b π b (i) ,...,t b π b (n b ) ≥ t b π b (n b −1) ), where the execution times (t b π b (1) ,...,t b π b (i) ) have been determined at lower depths. Consider any node (t b π b (1) ,...,t b π b (i) ,t b π b (i+1) ≥ t b π b (i) ,...,t b π b (n b ) ≥ t b π b (n b −1) ) at depthi. We now show how the auctioneer deter- minestheexecutiontimet b π b (i+1) atdepthi+1: Assumethatrobotrdoesnotwait forcomplextaskx b π b (i+1) inthegivenscenario. First,theauctioneerconstructsthe 168 (t 1 ≥ 0, t 2 ≥ t 1 , t 3 ≥ t 2 ) (t 1 = 7, t 2 ≥ t 1 , t 3 ≥ t 2 ) t 3 -13 ≥ t 1 –7 ≥ 0 t 2 -10 ≥ t 1 –5 ≥ 0 (t 1 = 7, t 2 = 12, t 3 ≥ t 2 ) t 2 -16 ≥ t 1 –11 ≥ 0 t 2 -12 ≥ t 1 –5 ≥ 0 (t 1 = 7, t 2 = 12, t 3 ≥ t 2 ) (t 1 = 7, t 2 = 14, t 3 ≥ t 2 ) t 3 -9 ≥ t 2 –4 ≥ 0 (t 1 = 7, t 2 = 12, t 3 = 17) t 3 -9 ≥ t 2 –4 ≥ 0 (t 1 = 7, t 2 = 14, t 3 = 19) depth 0 r 2 determines t 2 r 3 determines t 3 r 1 determines t 1 depth 1 depth 2 depth 3 leaf nodes Figure 5.13(a) (t 1 = 7 is inconsistent with the constraint t 1 –11 ≥ 0) Figure 5.13(b) Figure 5.13(c) Figure5.12: SearchTreeforDeterminingOptimalVisitTimes(MiniMaxTeamObjec- tive) permutation π r = (x r π r (1) ,...,x r π r (nr) ) = f r (π b ) of the complex tasks allocated torobotr andfindsthecomplextaskx r π r (j) ∈ π r thatisidenticaltocomplextask x b π b (i+1) ∈ π b . Then, for each execution order p∈P r,π r , it determines the exe- cution time t r π r (j) of complex task x r π r (j) to be t r π r (j−1) +t p π r (j) −t p π r (j−1) , where t r π r (j−1) is the execution time of complex task x r π r (j−1) that has been determined in previous depths. Finally, the auctioneer generates one child node in which t b π b (i+1) = t r π r (j) and labels the edge from the node in question to the child node with the domain constraintB(t r π r−t p ) = true of meta-functionF r,p π r (t r π r). If thedeterminedexecutiontimesinthechildnodeareinconsistentwithanydomain constraint labeling the edges from the root to the child node, then the auctioneer removes the child node from the search tree. This completes the construction of the search tree. Finally, the auctioneer adds the vectors of execution times in the leafnodesofthesearchtreetosetT. Consideragainthemulti-robotroutingexampleshowninFigure5.8. Therearesix possiblepermutationsofthethreecomplextargets{x 1 ,x 2 ,x 3 }andeightpossible scenariossothateachcomplextargethasonerobotthatdoesnotwaitatit,astab- ulatedinTable5.4. Assumethattheauctioneerconsiderspermutation(x 1 ,x 2 ,x 3 ) and the scenario that robot r 1 does not wait at complex target x 1 , robot r 2 does not wait at complex target x 2 , and robot r 3 does not wait at complex target x 3 . 169 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 3 (7, 13) no wait at target x 1 t 1 = 7 Thereactionfunctionofrobotr 1 (a) 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 (7, 14) (7, 12) no wait at target x 2 (5, 10) (5, 12) (11, 16) Thereactionfunctionofrobotr 2 (b) 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 2 t 3 (12, 17) (14, 19) no wait at target x 3 (4, 9) Thereactionfunctionofrobotr 3 (c) Figure5.13: DomainBoundariesforDeterminingOptimalVisitTimes Figure5.12showsthesearchtreeconstructedbytheauctioneer. Therootnodeis (t 1 ≥ 0,t 2 ≥ t 1 ,t 3 ≥ t 2 ), which implies the visit order of complex targets to be permutation (x 1 ,x 2 ,x 3 ). Since the givenscenario specifies that robotr 1 does not wait at complex target x 1 , the visit time of target x 1 is thus determined by robot r 1 at depth 1 of the search tree. Figure 5.13(a) shows the permutation function of robot r 1 for the permutation (x 1 ,x 3 ) of its allocated complex targets, which contains only one meta-functionF r 1 ,r 1 x 1 x 3 (x 1 ,x 3 ) (t 1 ,t 3 ). Thus, auctioneer generates only one child node (t 1 = 7,t 2 ≥ t 1 ,t 3 ≥ t 2 ) and labels the edge from the root node to this child node with the domain constraint t 3 − 13≥ t 1 − 7≥ 0. Similarly, robot r 2 determines the visit time of complex target x 2 at depth 2. Since there are three meta-functions for the permutation function of robot r 2 , as shown in Figure 5.13(b), (whose domain constraints are t 2 − 10≥ t 1 − 5≥ 0, t 2 −12≥t 1 −5≥ 0,andt 2 −16≥t 1 −11≥ 0),theauctioneergeneratesthree child nodes. However, the visit times in child node (t 1 = 7,t 2 = 12,t 3 ≥ t 2 ) are inconsistent with the domain constraint t 2 − 16 ≥ t 1 − 11 ≥ 0 of meta- functionF r 2 ,r 2 x 4 x 1 x 2 (x 1 ,x 2 ) (t 1 ,t 2 ),andthustheauctioneerhastoremovethischild node. Finally, robot r 3 determines the visit time of complex target x 3 at depth 3 170 ast 3 =t 2 +9−4foreachconsistentchildnodeatdepth2,whichcompletesthe construction of the search tree. The auctioneer then adds the two vectors of visit times in the leaf nodes of the search tree, namely (t 1 = 7,t 2 = 12,t 3 = 17) and (t 1 = 7,t 2 = 14,t 3 = 19),tosetT. Now we calculate an upper bound on the cardinality of set T. Given a bundle of relatedrobotsbandn b relatedcomplextasksallocatedtob,letn d bethelargestcoalition size of any complex task andn P be the largest number of different execution orders of anyrobotr∈btoexecuteitsallocatedtasks. First,thereareatmostn P meta-functions in the generalized reaction function of each robotr∈ b for its allocated complex tasks since each meta-function corresponds to one possible execution order of the allocated tasks for robotr. Second, there are at mostn b ! different permutations of complex tasks andn d n b different scenarios so that each complex task has one robot that does not wait for it. Third, the number of leaf nodes in the search tree constructed by the auctioneer foreachcombinationofapermutationandascenarioisatmostn P n b sincethedepthof thesearch tree isn b andeach node has atmostn P childnodes. Thus, the cardinality of setT isboundedbyn b !·(n d ·n P ) n b ,whichisafinitenumber. ApproximatingGeneralizedReactionFunctions Although the number of meta-functions in the (generalized) reaction function of each robot r ∈ b is finite, it is still exponential in the number of its allocated tasks since the number of total execution orders of robotr is exponential in the number of its allo- catedtasks. Inordertomakethecomputationandcommunicationofreactionfunctions efficient, we now show how to approximate reaction functions by considering only a constantnumberofexecutionordersofrobotr,whichissimilartotheideausedincase ofdisjointcoalitionsasdescribedinSubsection5.3.2. 171 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 (5, 12) (11, 16) (5, 10) (9, 4) (11, 4) (15,10) Thedomainoftheideal(generalized) reactionfunction 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 (5, 12) (5, 10) (9, 4) (11, 4) Thedomainoftheapproximated reactionfunctionwithG = 5 2 4 6 8 10 12 0 14 16 18 20 12 8 10 6 4 2 14 18 16 20 t 1 t 2 Thedomainoftheapproximated reactionfunctionwithG = 10 Figure 5.14: Approximating Generalized Reaction Function of Robot r 2 (MiniMax TeamObjective) 1. Robot r finds a sufficiently large execution time e so that it can execute all allo- cated(simpleandcomplex)tasksinanyexecutionorder(withoutwaitingforany task) before time e. For example, ∑ x∈X r s ∪X r c (d(r,x) +d(x,r)) +ϵ is a simple choice of e in a multi-robot routing problem, where d(r,x) is the travel time of robotrfromitsinitiallocationtoanytargetx∈X r s ∪X r c ,d(x,r)isthetraveltime of robot r from any target x back to its initial location, and ϵ is a small positive constant. Robotr thendividesthetimeinterval [0,e)evenlyintoG timeintervals [s j ,e j )foragivendiscretizationgranularityG. Finally,robotr constructsallpos- siblevectorsofn r timeintervals([s 1 ,e 1 ),...,[s nr ,e nr )),whereeach [s i ,e i )with 1≤i≤n r inthevectorcanbeany [s j ,e j )with 1≤j≤G. 2. Robotr solves the following optimization problem for each vector of time inter- vals ([s 1 ,e 1 ),...,[s nr ,e nr )), namely, to determine the execution order p of its tasksX r s ∪X r c withthesmallestrobotcostsuchthat: 1)robotr executescomplex taskx r i atsometimet p i ∈ [s i ,e i )and2)itdoesnotwaitforanytask. (Suchanexe- cution order does not always exist.) This optimization problem is a special case of job scheduling problem with time windows [CL87, LRS04]. We can still use 172 the Or-opt algorithm shown in Figure 5.6 to solve it approximately 6 . Let the set P r bethesetofexecutionorderspfoundforallpossiblevectorsoftimeintervals ([s 1 ,e 1 ),...,[s nr ,e nr )). 3. Theremainingstepsofrobotr forconstructingitsapproximategeneralizedreac- tion function are identical to Steps 2-6 of the procedure for constructing general- izedreactionfunctionsifonereplacesP r with P r everywhere. Clearly, P r ⊆P r and P r =P r if robotr uses an infinite number of time intervals (G =∞)andsolvestheoptimizationproblemsinStep2optimally. Ingeneral,approx- imatereactionfunctionsmaynotbeidenticaltotheidealreactionfunctions. Forexam- ple, Figure 5.14 shows the domains of the meta-functions of two approximate reaction functions obtained withe = 20 +ϵ and different discretization granularities. Consider the discretization granularityG = 5, as shown in the center of Figure 5.14. For the vectoroftimeintervals([4+ ϵ 5 ,8+ 2ϵ 5 ),[8+ 2ϵ 5 ,12+ 3ϵ 5 ))ofcomplextargetsx 1 andx 2 , there are two visit orders of allocated targets for robot r 2 that satisfy the constraints of thisvectoroftimeintervalsdescribedinStep2ofourapproximationapproach,namely (r 2 x 1 x 2 x 4 )(Order1)withvisittimes(5,10)ofcomplextargets(x 1 ,x 2 )and robot cost 13, and (r 2 x 1 x 4 x 2 ) (Order 2) with visit times (5,12) of complex targets (x 1 ,x 2 )androbotcost 12. SincetherobotcostofOrder2issmallerthanthatof Order1,theapproximatedreactionfunctionofrobotr 2 withG = 5ande = 20+ϵdoes nottakeintoaccountthemeta-functionforvisitorder(r 2 x 1 x 2 x 4 ). Similarly, 6 TheonlynecessarymodificationisonfunctionEvaluate(s): Forthecaseofoverlappingcoalitions, Evaluate(s) has to evaluate all possible execution orders of tasks in which simple tasks are executed in orders. Foreachsuchexecutionorder,letx r betheorderinwhichcomplextasksareexecutedbyrobots in this execution order. Then, it calculates the resulting robot cost of robot r to execute all tasks in this execution order without waiting at any task. It then returns the triple of the robot cost c, the execution timest r ofthecomplextasksx r ,andtheexecutionorderp,minimizedoverallpossibleexecutionorders (if any) where each execution time t r i ∈ t r of complex task x r i ∈ x r falls into time interval [s i ;e i ). If therearenosuchcases,itreturnsinfinityfortherobotcost,infinityfortheexecutiontimesandNULLfor theexecutionorder. 173 the visit order (r 2 x 2 x 4 x 1 ) dominates the visit order (r 2 x 2 x 1 x 4 ) for the vector of time intervals ([8+ 2ϵ 5 ,12+ 3ϵ 5 ),[0,4+ ϵ 5 )) of complex targetsx 1 and x 2 , and thus the approximated reaction function of robotr 2 withG = 5 ande = 20+ϵ does not contain the meta-function for visit order (r 2 x 2 x 1 x 4 ). Consider the discretization granularityG = 10, as shown in the right part of Figure 5.14. Since eachvectoroftimeintervalsofcomplextargetsx 1 andx 2 coversatmostonevisitorder of the targets allocated to robotr 2 , the approximated reaction function of robotr 2 with G = 10ande = 20+ϵisidenticaltotheidealreactionfunctionofrobotr 2 . Thenumberofpossiblevectorsoftimeintervals([s 1 ,e 1 ),...,[s nr ,e nr ))isbounded byG nr (whilestillexponentialinn r ),wheren r isthenumberofcomplextasksallocated torobotr. Thus,thenumberofexecutionordersin P r andthusalsothenumberofmeta- functionsintheapproximatereactionfunctionofrobotr isatmostG nr . Thecardinality ofthesetT usedbytheauctioneertodeterminetheoptimalexecutiontimesisbounded byn b !·(n d ·G nr ) n b ,wheren d isthelargestcoalitionsizeofthecomplextasksallocated to the bundleb of robots related to robotr andn b is the total number of these complex tasks. Given the assumptions discussed in Subsection 5.4.1, we have n d ≤D (from Assumption A), n r ≤C (from Assumption B ′ ). Thus, the size of the approximate reaction function is constant but the number of possible execution times the auctioneer hastoevaluatebasedontheapproximatereactionfunctionsisstillexponentialinn b . 5.4.3 SequentialAuctionswithGeneralizedReactionFunctions Nowweshowthatsequentialauctionswithreactionfunctions,asdevelopedinSubsec- tion 5.3.3, can be extended to the case of overlapping coalitions based on the usage of generalized reaction functions. We call the new auction algorithm sequential auctions withgeneralizedreactionfunctions. 174 Asequentialauctionwithgeneralizedreactionfunctionsconsistsofmultiplerounds to allocate tasks to robots. Initially, all tasks are unallocated. During each round, all robotsbidoneachunallocated(simpleorcomplex)taskageneralizedreactionfunction and then the auctioneer allocates one additional task to a robot or a coalition of robots so that the team cost increases the least. Consider any round of the sequential auction with generalized reaction functions. Assume that the current allocation of each robot r∈Ris(X r s ,X r c ,C r ),whereX r s isthesetofsimpletasksandX r c isthesetofcomplex tasksthathavealreadybeenallocatedtorobotrinpreviousrounds. LetX r =X r s ∪X r c . Then,U =X\ ∪ r∈R X r isthesetofunallocatedtasks. Wenowexplainexactlywhich bidsrobotssubmitandhowtheauctioneermakestheallocationdecision. • Bid-Construction Phase: Similar to the case of disjoint coalitions, robotr∈ R considers each taskx∈ U: Let ~ X r c be the set of complex tasks allocated to robot r if we tentatively allocate taskx to robotr,x r be the vector of complex tasks in ~ X r c , and t r be the corresponding vector of execution times of complex tasks x r . Ifrobotr iseligiblefortaskx,thenitconstructsa(generalized)reactionfunction F r,x ~ X r c (t r )asfollows: –F r,x ~ X r c (t r ) = F r,x X r c ∪{x} (t r ) = c robot r (X r s ,X r c ∪{x},{x r ← t r } if task x is complex; –F r,x ~ X r c (t r ) =F r,x X r c (t r ) =c robot r (X r s ∪{x s },X r c ,{x r ←t r })iftaskxissimple andX r c ̸=∅. –F r,x ~ X r c (t r ) =F r,x X r c (t r ) =c robot r (X r s ∪{x s },∅,∅)iftaskxissimpleandX r c =∅. 7 andbidsthisreactionfunctionF r,x ~ X r c (t r )ontaskx. 7 AlthoughF r;x ∅ (t r ) = c robot r (X r s ∪{x s };∅;∅) is a numerical value, we still write it as a (trivial) constantfunctionofvectorsofexecutiontimesforthesakeofconvenience. 175 • Winner-DeterminationPhase: Similartothecaseofdisjointcoalitions,theauc- tioneerkeepsthefollowinginformationforeachrobotr∈R: 1)itscurrentalloca- tion (X r s ,X r c ,C r ), 2) its current robot costc robot r (X r s ,X r c ,C r ), and 3) its reaction functionF r X r c (t r ) = c robot r (X r s ,X r c ,{x r ← t r }) if robotr has been allocated one ormorecomplextasksinpreviousrounds. If the auctioneer tentatively allocates task x to coalition P, let b(x,P)⊆ R be thebundleofrobotsthatarerelatedtotherobotsincoalitionP andx b(x,P) bethe vector of the related complex tasks for bundleb(x,P). An allocation of taskx to coalitionP islegaliffthenumberofrelatedrobotsinbundleb(x,P)isnotlarger thanR, as required by Assumption C in Subsection 5.4.1. Now we show how theauctioneerallocatesoneadditionaltaskinU torobotssothatthecurrentteam costincreasestheleast. – Solutionofthefirstsubproblem: For each unallocated simple or complex task x ∈ U, the auctioneer first constructs the set P(d(x)) that contains all qualified coalitions ofd(x) different eligible robot(s) to which it is legal to allocate task x. For each coalition P ∈ P(d(x)), the auctioneer then determines the evaluation cost c eval x,P of the allocation of task x to coalition P and the optimal execution times t b(x,P) of the related complex tasks in x b(x,P) (if the set of the related complex tasks in x b(x,P) is not empty) as described by below in the solution of the second subproblem. After that, the auctioneer determines (x,P) := argmin x∈U,P∈P(d(x)) c eval x,P . Finally, the auctioneer allocates taskx to all robots inP and sets the execution times of the related complex tasks x b(x,P) (if any) to t b(x,P) for all related robots in b(x,P). 176 – Solution of the second subproblem: For each legal allocation of taskx to coalitionP,theauctioneerfirstconstructsthefollowingevaluationfunction foreachrobotr∈b(x,P): V r X r c (t r ) :=    F r X r c (t r )−c robot r (X r s ,X r c ,C r ) (MiniSumandMiniLat) F r X r c (t r ) (MiniMax). ThereactionfunctionF r X r c (t r )ofrobotr isavailabletotheauctioneer: F r X r c (t r ) :=    F r,x ~ X r c (t r ) ifr∈P F r X r c (t r ) ifr / ∈P. whereF r,x ~ X r c (t r )isthereactionfunctionsubmittedbyrobotr ontaskxinthe bid-constructionphaseandF r X r c (t r )isthereactionfunctionofrobotrforits currentlyallocatedcomplextasksX r c . Finally,theauctioneerdetermines c eval x,P = min t b(x;P) ∈R n b(x;P) + c team r∈b(x,P) V r X r c (f r (t b(x,P) )) astheevaluationcostoftheallocationoftaskxtocoalitionP and t b(x,P) = arg min t b(x;P) ∈R n b(x;P) + c team r∈b(x,P) V r X r c (f r (t b(x,P) )) as the execution times of the related complex tasks in x b(x,P) if the set of relatedcomplextasksinx b(x,P) isnotempty 8 . 8 Ifthesetoftherelatedcomplextasksinx b(x;P) isempty,thenwehave b(x;P) ={r}and X r c =∅. In the bid-construction phase, robot r bids a constant functionF r;x ∅ (t r ) = c robot r (X r s ∪{x s };∅;∅) on task x. Thus, the evaluation cost of the allocation of task x to robot r can be still determined as the evaluationfunctionV r ∅ (t r )ofrobotr,namelyc robot r (X r s ∪{x s };∅;∅)−c robot r (X r s ;∅;∅)fortheMiniSum andMiniLatteamobjectivesandc robot r (X r s ∪{x s };∅;∅)fortheMiniMaxteamobjective. 177 The following theorem shows that the allocation decisions of sequential auctions with generalized reaction functions achieve the performance principle of complex-task auctions. Theorem5.4.3 Allocating task x to all d(x) robots r ∈ P with the execution times t b(x,P) for the related complex tasks x b(x,P) (if any) of the robots in bundle b(x,P) as determined by the auctioneer results in the smallest team cost among all eligible possi- bleallocationsofanyunallocatedtaskx ′ ∈U toanyqualifiedcoalitionofd(x ′ )eligible robots. Proof: The proof is identical to that of Theorem 5.3.4 for the case of disjoint coali- tions except that the set of related robots here isb(x,P) and the minimization to calcu- late the evaluation cost here is over a vector of execution times instead of a single one. 5.5 ExperimentalEvaluation Wenowevaluatetheperformanceofsequentialauctionswithreactionfunctionsempiri- callyforsolvingcomplex-taskallocationproblemsinthedomainofmulti-robotrouting. 5.5.1 ExperimentalSettings We evaluate our task-allocation algorithms for solving multi-robot routing problems withthefollowingdifferentcoalitiontypesinourexperiments: • DisjointCoalitions: Eachrobotcanvisitatmostonecomplextarget(thatis, the complex-targetcapacityofallrobotsisone). • OverlappingCoalitions: Eachrobotcanvisitatmostthreecomplextargets(that is,thecomplex-targetcapacityofallrobotsisthree). 178 Wecomparethefollowingcomplex-taskallocationalgorithms: 9 • SK Auctions: This algorithm implements the greedy auction algorithm as described in Section 5.2 where robots visit their targets in the same order as they are allocated to them. SK Auctions can solve multi-robot routing problems with bothcoalitiontypes. • SARF (Simple-First): When solving multi-robot routing problems with disjoint coalitions,thisalgorithmimplementssequentialauctionswithreactionfunctions, as described in Subsection 5.3.3, with the restriction that all simple targets are allocated to robots before any complex target. When solving multi-robot routing problems with overlapping coalitions, this algorithm implements sequential auc- tions with generalized reaction functions, as described in Subsection 5.4.3, with thesamerestriction. • SARF (Complex-First): When solving multi-robot routing problems with dis- jointcoalitions,thisalgorithmimplementssequentialauctionswithreactionfunc- tions,asdescribedinSubsection5.3.3,withtherestrictionthatallcomplextargets areallocatedtorobotsbeforeanysimpletarget. Whensolvingmulti-robotrouting problems with overlapping coalitions, this algorithm implements sequential auc- tions with generalized reaction functions as described in Subsection 5.4.3, with thesamerestriction. • SARF (Mixed): When solving multi-robot routing problems with disjoint coali- tions, this algorithm implements sequential auctions with reaction functions, as described in Subsection 5.3.3, without any restriction. When solving multi-robot 9 We also attempt to find the smallest team cost by formulating multi-robot routing with complex targets as mixed integer programs. We seed the solutions with the allocations produced by sequential auctions with reaction functions and then solve mixed integer programs with CPLEX. However, CPLEX wasunabletoreducetheteamcostsoftheinitialsolutionswithinatimelimitofthreehours. 179 routing problems with overlapping coalitions, this algorithm implements sequen- tialauctionswithgeneralizedreactionfunctions,asdescribedinSubsection5.4.3, withoutanyrestriction. • Random: This algorithm implements a randomized task-allocation algorithm whereeachtargetxisrandomlyallocatedtod(x)eligiblerobots. We evaluate our task-allocation algorithms on three different kinds of terrain, as showninFigure3.10. Inourexperiments,thereexistsonlyonecapability,asinprevious chapters. Robotsprovideoneunitofthiscapabilityeach,andsimpletargetsrequireone unitofthiscapabilityeach. However,complextargetsrequiretwounitsofthiscapability each. 10 In our experiments, we evaluate our task-allocation algorithms with the following robotsimple-targetcapacityconstraints: • TightRobotSimple-TargetCapacityConstraint: Thesumoftherobotsimple- target capacities of all robots is equal to the number of simple targets. Unless explicitlymentionedotherwise,therobotsimple-targetcapacity-constraintistight inourexperiments. • LooseRobotSimple-TargetCapacityConstraint: Thesumoftherobotsimple- targetcapacitiesofallrobotsisgreaterthanthenumberofsimpletargets. We evaluate sequential auctions with reaction functions with the following approachesofreaction-functioncalculations: 10 Theexperimentalsettingshereassumethatthereexistsonlyonecapabilityforbothrobotsandtasks, which is restrictive on how complex tasks become related. It is future work to design more sophisticated experimentalsettingsforbetterevaluatingourcomplex-taskauctionalgorithms. 180 • ApproximatedReaction-FunctionCalculation: Ifarobotdoesnotgetallocated any complex targets, then it uses the two-opt heuristic to calculate the approx- imated robot cost of visiting its allocated simple targets. Otherwise, the robot usestheOr-optheuristicasdescribedinSubsection5.3.2toconstructthereaction functionforitsallocatedcomplextargetinmulti-robotroutingwithdisjointcoali- tions, and uses the Or-opt heuristic as described in Subsection 5.4.2 to construct the generalized reaction function for its allocated complex targets in multi-robot routing with overlapping coalitions. The discretization granularityG is 30 and ϵ is 1. Unless explicitly mentioned otherwise, the reaction-function calculation is approximatedinourexperiments. • Ideal Reaction-Function Calculation: A robot always evaluates all possible routesofvisitingitsallocatedtargetstocalculateitsrobotcostsandreactionfunc- tions. 5.5.2 ExperimentalResults Weperformexperimentswithdifferentsettingsintroducedintheprevioussubsectionto evaluateourcomplex-taskauctionalgorithms,assummarizedinTable5.5. Thedetailed experimental data and explanations can be found in Appendex A.3. We make the fol- lowingobservations: TeamCosts • The team costs of sequential auctions with reaction functions are smaller than those of SK Auctions that do not use reaction functions. This observation meets our expectation since robots can update the visit times of their allocated targets 181 Results Coalition Terrain Blockage Simple-Target Robots Simple Complex Reaction-Function RobotSimple-Target Type Status Capacities Targets Targets Calculations CapacityConstraint TableA.42 TeamCosts Disjoint Outdoor-Like Known 3 2-10 6-30 1-5 Approximated Tight TableA.43 ResponseTimes Disjoint Outdoor-Like Known 3 2-10 6-30 1-5 Approximated Tight TableA.44 TeamCosts Disjoint Empty Known 3 2-10 6-30 1-5 Approximated Tight TableA.45 ResponseTimes Disjoint Empty Known 3 2-10 6-30 1-5 Approximated Tight TableA.46 TeamCosts Disjoint Indoor-Like Known 3 2-10 6-30 1-5 Approximated Tight TableA.47 ResponseTimes Disjoint Indoor-Like Known 3 2-10 6-30 1-5 Approximated Tight TableA.48 TeamCosts Disjoint Outdoor-Like Known 2-5 10 20-50 5 Approximated Tight TableA.49 ResponseTimes Disjoint Outdoor-Like Known 2-5 10 20-50 5 Approximated Tight TableA.50 TeamCosts Disjoint Outdoor-Like Known 2-5 10 20-50 5 Ideal Tight TableA.51 ResponseTimes Disjoint Outdoor-Like Known 2-5 10 20-50 5 Ideal Tight TableA.52 TeamCosts Disjoint Outdoor-Like Known 3-∞ 10 30 5 Approximated Loose TableA.53 ResponseTimes Disjoint Outdoor-Like Known 3-∞ 10 30 5 Approximated Loose TableA.54 TeamCosts Disjoint Outdoor-Like Unknown 3 2-10 6-30 1-5 Approximated Tight TableA.55 RepeatedExecutions Disjoint Outdoor-Like Unknown 3 2-10 6-30 1-5 Approximated Tight TableA.56 ResponseTimes Disjoint Outdoor-Like Unknown 3 2-10 6-30 1-5 Approximated Tight TableA.57 TeamCosts Overlapping Outdoor-Like Known 3 2-10 6-30 3 Approximated Tight TableA.58 ResponseTimes Overlapping Outdoor-Like Known 3 2-10 6-30 3 Approximated Tight TableA.59 TeamCosts Overlapping Empty Known 3 2-10 6-30 3 Approximated Tight TableA.60 ResponseTimes Overlapping Empty Known 3 2-10 6-30 3 Approximated Tight TableA.61 TeamCosts Overlapping Indoor-Like Known 3 2-10 6-30 3 Approximated Tight TableA.62 ResponseTimes Overlapping Indoor-Like Known 3 2-10 6-30 3 Approximated Tight TableA.63 TeamCosts Overlapping Outdoor-Like Known 2-5 10 20-50 3 Approximated Tight TableA.64 ResponseTimes Overlapping Outdoor-Like Known 2-5 10 20-50 3 Approximated Tight TableA.65 TeamCosts Overlapping Outdoor-Like Known 2-5 10 20-50 3 Ideal Tight TableA.66 ResponseTimes Overlapping Outdoor-Like Known 2-5 10 20-50 3 Ideal Tight TableA.67 TeamCosts Overlapping Outdoor-Like Known 3-∞ 10 30 3 Approximated Loose TableA.68 ResponseTimes Overlapping Outdoor-Like Known 3-∞ 10 30 3 Approximated Loose TableA.69 TeamCosts Overlapping Outdoor-Like Known 3 10 30 3-7 Approximated Tight TableA.70 ResponseTimes Overlapping Outdoor-Like Known 3 10 30 3-7 Approximated Tight TableA.71 TeamCosts Overlapping Outdoor-Like Unknown 3 2-10 6-30 3 Approximated Tight TableA.72 RepeatedExecutions Overlapping Outdoor-Like Unknown 3 2-10 6-30 3 Approximated Tight TableA.73 ResponseTimes Overlapping Outdoor-Like Unknown 3 2-10 6-30 3 Approximated Tight Table5.5: SummaryofComplex-TaskAuctionExperiments when a new target is allocated to them in sequential auctions with reaction func- tions. • TheteamcostsoftheSNW(Mixed)algorithmaresmallerthanthoseoftheSNW (Simple-First,Complex-First)algorithms. Thisobservationmeetsourexpectation since there are more constraints in allocating targets in the SNW (Simple-First, Complex-First)algorithms. ResponseTimes • The response times of sequential auctions with reaction functions are larger than those of SK Auctions. This observation meets our expectation since robots sequential auctions with reaction functions need to construct reaction functions whentheyareallocatedanycomplextargets. 182 • Theresponsetimesofsequentialauctionswithreactionfunctionsaresmallwhen thenumberofrelatedcomplextargetsissmall(lessthan5). • The response times of sequential auctions with generalized reaction functions increasedramaticallywiththenumberofcomplextargets. Thisobservationmeets our expectation since the runtime of determining visit times of related complex targetsisexponentialinthenumberofrelatedcomplextargets. 5.6 Summary Inthischapter,westudiedauctionalgorithmsforsolvingcomplex-taskallocationprob- lems. We developed a new auction algorithm, called sequential auctions with reaction functions, that allocates a complex or simple task to robots in each round so that the team cost increases the least. Our theoretical results show that reaction functions char- acterizethenecessarylocalinformationofthebiddersneededbytheauctioneertomake the allocation decisions so that the team cost increases the least. We also developed approximationtechniquesforconstructingreactionfunctionssothatthenumberofseg- ments in a reaction function is constant. Our experimental results show that sequential auctionswithreactionfunctionsreducetheteamcostsofSKauctionswhilemaintaining theadvantageofsmallresponsetimes(whenthenumberofcomplextargetsissmall). 183 Chapter6 Complex-TaskNegotiations Inthischapter,westudytask-allocationalgorithmsthatimplementtheone-to-onenego- tiationprotocolshowninFigure1.8forsolvingcomplex-taskallocationproblems. The auction algorithms studied in the previous chapter do not re-allocate tasks once they have been allocated to robots. In this chapter, we study how robots can negotiate with each other to exchange their allocated tasks in order to improve their current alloca- tions. We have demonstrated in Chapter 4 that sequential negotiations with K-swaps, the negotiation algorithm that re-allocates simple tasks among robots, can significantly reduce the team cost of sequential auctions with bundles for solving simple-task allo- cation problems. Now, we develop similar negotiation algorithms that re-allocate both simpleandcomplextasksamongrobots. In this chapter, we illustrate that one-to-one task-swaps [GMR97a], which were originally developed to exchange two simple tasks between two robots in each round in the context of simple-task allocation problems, can be extended to exchange either two simple or two complex tasks between two robots in each round in the context of complex-task allocation problems. Our goal is to develop negotiation algorithms for solving complex-task allocation problems (complex-task negotiations) that achieve the followingmyopicoptimizationperformanceprincipleonsingletasks: Performance principle of complex-task negotiations: To perform a simple-taskorcomplex-taskexchangebetweentworobotsineachroundof complex-tasknegotiationssothatthecurrentteamcostdecreasesthemost. 184 We proceed as follows: Section 6.1 formalizes the problem studied in this chapter. Section 6.2 introduces a new negotiation algorithm, called sequential negotiations with reactionfunctions,inwhichrobotsre-allocate(eithercomplexorsimple)tasksbetween themineachroundsothattheteamcostdecreasesthemostbyusingreactionfunctions as introduced in the previous chapter. Robots in Section 6.2 are assumed to be able to execute at most one complex task. Section 6.3 relaxes this assumption. Finally, Section6.4presentsexperimentalresults,andSection6.5concludesthischapter. 6.1 ProblemDescription We study complex-task allocation problems as formalized in Section 5.1. Let ((X r s ,X r c ,C r )) r∈R = ((X r 1 s ,X r 1 c ,C r 1 ),...,(X rm s ,X rm c ,C rm )) be a solution of a given complex-task allocation problem⟨R,X,E⟩. In this chapter, we study how two robots can exchange their allocated simple or complex tasks to produce a new solution (( X r s , X r c , C r )) r∈R = (( X r 1 s , X r 1 c , C r 1 ),...,( X rm s , X rm c , C rm )) so that the team cost of the new allocations (( X r s , X r c , C r )) r∈R is smaller than that of the initial allocations ((X r s ,X r c ,C r )) r∈R . Before we proceed to describe our negotiation algorithms, we first formalize which types of task exchanges robots can perform. A task exchange (r,r ′ ,x,x ′ ) between two differentrobotsr,r ′ ∈Rdescribesthatrobotrmovesitsallocatedtaskxtorobotr ′ and robotr ′ movesitsallocatedtaskx ′ torobotr. Atmostoneofthetasksx orx ′ inatask exchangecanbeempty,writtenas∅. Ataskexchange(r,r ′ ,x,x ′ )betweenrobotsr and r ′ is legal iff 1) (x∈ X r s andx ′ ∈ X r ′ s ) or (x∈ X r c andx ′ ∈ X r ′ c ), that is, it is either a simple-taskexchangeoracomplex-taskexchange; 2)E r = E r ′ ,thatis, robotsr andr ′ provide the same capability; and 3) Ifx =∅,|X r s ∪{x ′ }|≤ q r s if taskx is simple, and 185 r 3 r 1 r 2 x 1 x 2 x 3 r 4 Figure6.1: Example12ofMulti-RobotRouting r 3 r 1 r 2 x 1 x 3 r 4 x 2 OptimalAllocations (TeamCost=6) r 3 r 1 r 2 x 1 x 3 r 4 x 2 AllocationsofSequentialAuctionswithReactionFunctions (TeamCost=12) Figure6.2: AllocationsofExampleinFigure6.1(MiniMaxTeamObjective) |X r c ∪{x ′ }|≤ q r c if taskx ′ is complex; Ifx ′ =∅,|X r ′ s ∪{x}|≤ q r ′ s if taskx is simple, and|X r ′ c ∪{x}|≤q r ′ c iftaskxiscomplex. In the following, we study how two robots can construct all legal task exchanges betweenthemthateachreducetheteamcostoftheinitialallocations. 6.2 DisjointCoalitions We first consider complex-task allocation problems with disjoint coalitions. Figure 6.1 shows an example of multi-robot routing with complex targets and disjoint coalitions for the MiniMax team objective. It consists of four robots, one simple target, and two complex targets. There exists only one capability. Robotsr 1 ,r 2 ,r 3 andr 4 can provide one unit of this capability each. Target x 1 requires one unit of this capability, while 186 BidConstruction WinnerDetermination Bids(calculatedandsubmitted) Allocations(evaluated) Decision Round1 Robotr 1 bids:7ontargetx 1 (x 1 →r 1 ): 7 reactionfunction(Figure6.3(a))ontargetx 2 (x 1 →r 2 ): 5 reactionfunction(Figure6.3(b))ontargetx 3 (x 1 →r 3 ): 3 (x 1 →r 4 ): 4 Robotr 2 bids:5ontargetx 1 (x 2 →{r 1 ,r 2 }): 6 Theauctioneer reactionfunction(Figure6.4(a))ontargetx 2 (x 2 →{r 1 ,r 3 }): 6 allocatestarget reactionfunction(Figure6.4(b))ontargetx 3 (x 2 →{r 1 ,r 4 }): 6 x 1 torobotr 3 (x 2 →{r 2 ,r 3 }): 4 Robotr 3 bids:3ontargetx 1 (x 2 →{r 2 ,r 4 }): 5 reactionfunction(Figure6.5(a))ontargetx 2 (x 2 →{r 3 ,r 4 }): 5 reactionfunction(Figure6.5(b))ontargetx 3 (x 3 →{r 1 ,r 2 }): 12 (x 3 →{r 1 ,r 3 }): 12 Robotr 4 bids:4ontargetx 1 (x 3 →{r 1 ,r 4 }): 12 reactionfunction(Figure6.6(a))ontargetx 2 (x 3 →{r 2 ,r 3 }): 10 reactionfunction(Figure6.6(b))ontargetx 3 (x 3 →{r 2 ,r 4 }): 10 (x 3 →{r 3 ,r 4 }): 6 Round2 Robotr 1 bids:reactionfunction(Figure6.3(a))ontargetx 2 (x 2 →{r 1 ,r 2 }): 6 reactionfunction(Figure6.3(b))ontargetx 3 (x 2 →{r 1 ,r 3 }): 6 (x 2 →{r 1 ,r 4 }): 6 Robotr 2 bids:reactionfunction(Figure6.4(a))ontargetx 2 (x 2 →{r 2 ,r 3 }): 4 reactionfunction(Figure6.4(b))ontargetx 3 (x 2 →{r 2 ,r 4 }): 5 Theauctioneer (x 2 →{r 3 ,r 4 }): 5 allocatestarget Robotr 3 bids:reactionfunction(Figure6.5(a))ontargetx 2 (x 3 →{r 1 ,r 2 }): 12 x 2 torobotsr 2 reactionfunction(Figure6.5(c))ontargetx 3 (x 3 →{r 1 ,r 3 }): 12 andr 3 (x 3 →{r 1 ,r 4 }): 12 Robotr 4 bids:reactionfunction(Figure6.6(a))ontargetx 2 (x 3 →{r 2 ,r 3 }): 10 reactionfunction(Figure6.6(b))ontargetx 3 (x 3 →{r 2 ,r 4 }): 10 (x 3 →{r 3 ,r 4 }): 8 Round3 Robotr 1 bids:reactionfunction(Figure6.3(b))ontargetx 3 Theauctioneer (x 3 →{r 1 ,r 4 }): 12 allocatestarget Robotr 4 bids:reactionfunction(Figure6.6(b))ontargetx 3 x 3 torobotsr 1 andr 4 Table6.1: SequentialAuctionswithReactionFunctionsforExampleinFigure6.1(Min- iMaxTeamObjective) targetsx 2 and x 3 require two units. The simple-target capacity and the complex-target capacityofallrobotsarebothone. TheleftpartofFigure6.2showstheoptimalalloca- tions with the smallest team cost 6: First, robot r 4 arrives at complex target x 3 at time 3 and waits there until robot r 3 arrives at time 6. Second, robot r 2 visits simple target x 1 attime5andthenarrivesatcomplextargetx 2 attime6whenrobotr 1 alsoarrivesat complextargetx 2 . Finally,robotsr 1 andr 2 visitcomplextargetx 2 attime6,androbots r 3 and r 4 visit complex target x 3 at time 6. Table 6.1 tabulates the bids of robots and the decisions of the auctioneer in each round of sequential auctions with reaction func- tions as described in the previous chapter for solving this multi-robot routing example, wherereactionfunctionsareideallyconstructed. TherightpartofFigure6.2showsthe 187 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 2 robot cost (= task-completion time) (a) Reactionfunctionontargetx 2 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (b) Reactionfunctionontargetx 3 Figure6.3: ReactionFunctionsofRobotr 1 forSequentialAuctionswithReactionFunc- tionsforExampleinFigure6.1(MiniMaxTeamObjective) 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 2 robot cost (= task-completion time) (a) Reactionfunctionontargetx 2 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (b) Reactionfunctionontargetx 3 Figure6.4: ReactionFunctionsofRobotr 2 forSequentialAuctionswithReactionFunc- tionsforExampleinFigure6.1(MiniMaxTeamObjective) resulting allocations with team cost 12. Throughout this section, we use this example to illustrate how robots can construct legal complex-task and simple-task exchanges to improvetheirinitialallocationsproducedbysequentialauctionswithreactionfunctions. 6.2.1 Complex-TaskExchanges We first consider complex-task exchanges and develop a distributed negotiation proce- dureforconstructingalllegalcomplex-taskexchangesthateachreducetheteamcostof the current allocations ((X r s ,X r c ,C r )) r∈R . In the following, we explain exactly which proposalsrobotssendtoeachotherandhowtheyevaluateeachother’sproposals: 188 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 2 robot cost (= task-completion time) (a) Reactionfunctionontargetx 2 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (b) Reactionfunctionontargetx 3 (Round1) 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (c) Reactionfunctionontargetx 3 (Round2) Figure6.5: ReactionFunctionsofRobotr 3 forSequentialAuctionswithReactionFunc- tionsforExampleinFigure6.1(MiniMaxTeamObjective) 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 2 robot cost (= task-completion time) (a) Reactionfunctionontargetx 2 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (b) Reactionfunctionontargetx 3 Figure6.6: ReactionFunctionsofRobotr 4 forSequentialAuctionswithReactionFunc- tionsforExampleinFigure6.1(MiniMaxTeamObjective) • Task-Announcement Phase: Each robot r ∈ R sends the other robots the information about its allocated tasks in X r s and X r c , its current robot cost c r = c robot r (X r s ,X r c ,C r )anditsreactionfunctionforitscomplextaskinX r c (ifany). Consideragainthemulti-robotroutingexampleshowninFigure6.1fortheMin- iMax team objective. Robot r 1 sends out its currently allocated simple targets X r 1 s =∅ and complex targetX r 1 c ={x 3 }, its current robot cost 12, and its reac- tionfunctionfortargetx 3 showninFigure6.3(b). Robotr 2 sendsoutitscurrently allocated simple targets X r 2 s =∅ and complex target X r 2 c ={x 2 }, its current robotcost 4,anditsreactionfunctionfortargetx 2 showninFigure6.4(a). Robot 189 r 3 sends out its currently allocated simple targetX r 3 s ={x 1 } and complex target X r 3 c ={x 2 },itscurrentrobotcost 4,anditsreactionfunctionfortargetx 2 shown inFigure6.5(a). Robotr 4 sendsoutitscurrentlyallocatedsimpletargetsX r 4 s =∅ andcomplextargetX r 3 c ={x 3 },itscurrentrobotcost12,anditsreactionfunction fortargetx 3 showninFigure6.6(b). • Task-Exchange Evaluation Phase: Each robot r∈ R runs the following pro- cedure: If robot r is allocated no complex task, then it does nothing. Otherwise, robot r is currently allocated a complex task X r c ={x}. For each other robot r ′ ∈Rwithr̸=r ′ ,itexecutes: – Case 1: If robot r ′ is allocated a complex task X r ′ c = {x ′ } that is not allocated to robot r, then robot r considers the complex-task exchange S = (r,r ′ ,x,x ′ ) if it is legal. Let P x and P x ′ be the coalitions of related robots that are allocated to complex tasks x and x ′ in the current alloca- tions, respectively. Let R ′ = R\ (P x ∪ P x ′) be the set of robots that are notrelatedtorobotsrorr ′ beforeperformingthecomplex-taskexchangeS. Let P ′ x ′ = P x ′\{r ′ }∪{r} be the new coalition of related robots allocated to complex taskx ′ after performing the complex-task exchangeS. Robotr firstconstructsitsreactionfunctionF r x ′(t)forexecutingcomplextaskx ′ and thencalculatesitsnetlossofthecomplex-taskexchangeS as netloss(r,S) =          min 0≤t<∞ ∑ ~ r∈P ′ x ′ F ~ r x ′(t)− ∑ ~ r∈P ′ x ′ c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P ′ x ′ F ~ r x ′(t))−c (MiniMax). (6.1) 190 wherec =c team ~ r∈R c ~ r istheteamcostofthecurrentallocations(thatis,theteam cost before performing the complex-task exchange S). For the MiniSum and MiniLat team objectives,netloss(r,S) is the increase in the sum of the robot costs of all robots inP ′ x ′ after performing the complex-task exchange S. FortheMiniMaxteamobjective,netloss(r,S)isthedifferencebetween the largest robot cost of any robot in P ′ x ′∪R ′ and the current team cost c afterperformingthecomplex-taskexchangeS. – Case 2: If robotr ′ is allocated no complex task, then robotr considers the complex-task exchange S = (r,r ′ ,x,∅) if it is legal. Let R ′ = R\ (P x ∪ {r ′ }). Robotr calculatesitsnetlossofthecomplex-taskexchangeS as netloss(r,S) =      c robot r (X r s ,∅,∅)−c r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r ,c robot r (X r s ,∅,∅))−c (MiniMax). FortheMiniSumandMiniLatteamobjectives,netloss(r,S)istheincrease in the robot cost of robot r after performing the complex-task exchange S. FortheMiniMaxteamobjective,netloss(r,S)isthedifferencebetweenthe largest robot cost of any robot inR ′ ∪{r} and the current team costc after performingthecomplex-taskexchangeS. In either case, robotr proposes complex-task exchangeS = (r,r ′ ,x,x ′ ) to robot r ′ withvaluenetloss(r,S)bysending⟨S,netloss(r,S)⟩iffnetloss(r,S)≤ 0for theMiniSumandMiniLatteamobjectivesandnetloss(r,S)< 0fortheMiniMax teamobjective. Consideragainthemulti-robotroutingexampleshowninFigure6.1fortheMin- iMax team objective. Table 6.2 tabulates the complex-task exchanges evaluated 191 Robots Complex-TaskExchanges(evaluated) Complex-TaskExchanges(submitted) TaskExchanges Netloss ⟨TaskExchange,Value⟩ r 1 (r 1 ,r 2 ,x 3 ,x 2 ) -6 ⟨(r 1 ,r 2 ,x 3 ,x 2 ),−6⟩ (r 1 ,r 3 ,x 3 ,x 2 ) -6 ⟨(r 1 ,r 3 ,x 3 ,x 2 ),−6⟩ r 2 (r 2 ,r 1 ,x 2 ,x 3 ) -2 ⟨(r 2 ,r 1 ,x 2 ,x 3 ),−2⟩ (r 2 ,r 4 ,x 2 ,x 3 ) 0 r 3 (r 3 ,r 1 ,x 2 ,x 3 ) -4 ⟨(r 3 ,r 1 ,x 2 ,x 3 ),−4⟩ (r 3 ,r 4 ,x 2 ,x 3 ) 0 r 4 (r 4 ,r 2 ,x 3 ,x 3 ) -7 ⟨(r 4 ,r 2 ,x 3 ,x 3 ),−7⟩ (r 4 ,r 3 ,x 3 ,x 4 ) -7 ⟨(r 4 ,r 3 ,x 3 ,x 4 ),−7⟩ Table 6.2: Task-Exchange Evaluation Phase of Complex-Task Exchanges for Example inFigure6.1(MiniMaxTeamObjective) and proposed by each robot for the current allocations shown in the right part of Figure6.2. • Proposal-EvaluationPhase: Each robot r ∈ R runs the following procedure: If robot r does not receive any proposals, then it does nothing. Otherwise, for each proposal ⟨S = (r ′ ,r,x ′ ,x),netloss(r ′ ,S)⟩sentfromanotherrobotr ′ ,itexecutes: – Case1: Ifrobotr isallocatedacomplextask(thatis,X r c ={x}̸=∅),then it must have evaluated the complex-task exchange S ′ = (r,r ′ ,x,x ′ ) and calculated the corresponding netloss netloss(r,S ′ ) in the previous phase. Thecomplex-taskexchangeS ′ isidenticaltothecomplex-taskexchangeS. Thus, the netlossnetloss(r,S) of robotr for the complex-task exchangeS isequaltonetloss(r,S ′ ). – Case2: Ifrobotr isallocatednocomplextask(thatis,X r c =∅)thenitexe- cutes: LetR ′ =R\(P x ′∪{r}),andP ′ x ′ =P x ′\{r ′ }∪{r}bethenewcoali- tionofrobotsallocatedtocomplextaskx ′ afterperformingthecomplex-task exchangeS. Then,itconstructsthereactionfunctionF r x ′(t)forexecutingthe complextaskx ′ andthencalculatesitsnetlossofthecomplex-taskexchange S = (r ′ ,r,x ′ ,∅)withtheformulashowninFormula(6.1). 192 Robots Complex-TaskExchanges(received) Netloss Loss Profitable ⟨TaskExchange,Value⟩ Complex-TaskExchanges r 1 ⟨(r 2 ,r 1 ,x 2 ,x 3 ),−2⟩ -6 -2 (r 2 ,r 1 ,x 2 ,x 3 ) ⟨(r 3 ,r 1 ,x 2 ,x 3 ),−4⟩ -6 -4 (r 3 ,r 1 ,x 2 ,x 3 ) r 2 ⟨(r 1 ,r 2 ,x 3 ,x 2 ),−6⟩ -2 -2 (r 1 ,r 2 ,x 3 ,x 2 ) ⟨(r 4 ,r 2 ,x 3 ,x 2 ),−7⟩ 0 0 r 3 ⟨(r 1 ,r 3 ,x 3 ,x 2 ),−6⟩ -4 -4 (r 1 ,r 3 ,x 3 ,x 2 ) ⟨(r 4 ,r 3 ,x 3 ,x 2 ),−7⟩ 0 0 r 4 − − − Table6.3: Proposal-EvaluationPhaseofComplex-TaskExchangesforExampleinFig- ure6.1(MiniMaxTeamObjective) Finally,wedefinethelossofanycomplex-taskexchangeS = (r,r ′ ,x,x ′ )tobe loss(S) =c team (netloss(r,S),netloss(r ′ ,S)) andthendefinethecomplex-taskexchangeS tobeprofitableiffloss(S)< 0. Consideragainthemulti-robotroutingexampleshowninFigure6.1fortheMin- iMax team objective. Table 6.3 tabulates the proposals received from each robot andtheprofitablecomplex-taskexchanges. For any complex-task exchangeS = (r,r ′ ,x,x ′ ), complex tasksx andx ′ cannot be bothempty,therefore,weassumex̸=∅inthefollowing. Let ^ xbecomplextaskxorx ′ inalegalcomplex-taskexchangeS = (r,r ′ ,x,x ′ ),P ^ x bethecoalitionofrelatedrobots that are allocated to complex task ^ x before performing the complex-task exchange S (defineP x ′ ={r ′ } ifx ′ =∅), P ′ ^ x be the coalition of related robots that are allocated to complextask ^ xafterperformingthecomplex-taskexchangeS,R ′ =R\(P x ∪P x ′)be thesetrobotsthatarenotrelatedtorobotsr andr ′ beforeperformingthecomplex-task exchangeS,and(( X ~ r s , X ~ r c , C ~ r ) ~ r∈R bethenewallocationsafterperformingthecomplex- task exchange S on the current allocations ((X ~ r s ,X ~ r c ,C ~ r )) ~ r∈R . Now, we show that all complex-task exchanges that each reduce the team cost of the current allocations are evaluatedbyatleastonerobotintheproposal-evaluationphaseofthisprocedure. 193 Lemma6.2.1 For any legal complex-task exchange, we have c team ~ r∈P ′ ^ x c robot ~ r ( X ~ r s , X ~ r c , C ~ r )≥ min 0≤t<∞ c team ~ r∈P ′ ^ x F ~ r ~ x (t)if ^ x̸=∅. Proof: c team ~ r∈P ′ ^ x c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c team ~ r∈P ′ ^ x c robot ~ r (X ~ r s ,{^ x},{^ x← t}) ≥ min 0≤t<∞ c team ~ r∈P ′ ^ x c robot ~ r (X ~ r s ,{^ x},{^ x←t}) = min 0≤t<∞ c team ~ r∈P ′ ^ x F ~ r ^ x (t), given the definition of reaction functionF ~ r ^ x (t) = c robot ~ r (X ~ r s ,{^ x},{^ x← t}) for each robot ~ r∈P ′ ^ x . Lemma6.2.2 For any legal complex-task exchange S, we have c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )foreachrobot ~ r∈R ′ . Proof: The robot cost of each robot ~ r∈ R ′ does not change after performing the complex-taskexchangeS sincethecomplex-taskexchangeS doesnotchangetheallo- cation of robot ~ r : 1) the complex-task exchange S does not re-allocate any task for robot ~ r sinceitre-allocatesonlycomplextasksforrobotsr andr ′ ;and2)thecomplex- taskexchangeS doesnotchangethecommitment(ifany)ofrobot ~ r sinceitcanchange onlytheexecutiontimesforcomplextasksxandx ′ . Lemma6.2.3 Foranylegalcomplex-taskexchange,wehaveP x ∪P x ′ =P ′ x ∪P ′ x ′. Proof: Bydefinition,P ′ x =P x \{r}∪{r ′ }andP ′ x ′ =P x ′\{r ′ }∪{r}. Thus,P ′ x ∪P ′ x ′ = (P x \{r}∪{r ′ })∪(P x ′\{r ′ }∪{r}) =P x ∪P x ′. Theorem6.2.1 Foranylegalcomplex-taskexchangeS,wehave loss(S)≤c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). 194 Proof: ConsiderfirsttheMiniSumandMiniLatteamobjectives. Ifx ′ =∅,then loss(S) = netloss(r,S)+netloss(r ′ ,S) = c robot r (X r s ,∅,∅)−cr + min 0≤t<∞ ∑ ~ r∈P ′ x F ~ r x (t)− ∑ ~ r∈P ′ x c ~ r ≤ c robot r ( X r s ,∅,∅)+ ∑ ~ r∈P ′ x c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−(cr + ∑ ~ r∈P ′ x c ~ r ) (Lemma6.2.1and X r s =X r s ) = ∑ ~ r∈P ′ x ∪{r} c robot ~ r ( X ~ r s , X ~ r c , C ~ r )+ ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−( ∑ ~ r∈P ′ x ∪{r} c ~ r + ∑ ~ r∈R ′ c ~ r ) (Lemma6.2.2) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c ~ r (Lemma6.2.3andR ′ =R\(Px ∪P x ′)) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Ifx ′ ̸=∅,then loss(S) = netloss(r,S)+netloss(r ′ ,S) = min 0≤t<∞ ∑ ~ r∈P ′ x ′ F ~ r x ′(t)− ∑ ~ r∈P ′ x ′ c ~ r + min 0≤t<∞ ∑ ~ r∈P ′ x F ~ r x (t)− ∑ ~ r∈P ′ x c ~ r ≤ ∑ ~ r∈P ′ x c robot ~ r ( X ~ r s , X ~ r c , C ~ r )+ ∑ ~ r∈P ′ x ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−( ∑ ~ r∈P ′ x c ~ r + ∑ ~ r∈P ′ x ′ c ~ r ) (Lemma6.2.1) = ∑ ~ r∈P ′ x ∪P ′ x ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )+ ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−( ∑ ~ r∈P ′ x ∪P ′ x ′ c ~ r + ∑ ~ r∈R ′ c ~ r ) (Lemma6.2.2) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c ~ r (Lemma6.2.3andR ′ =R\(Px ∪P x ′)) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). . NowconsidertheMiniMaxteamobjective. Ifx ′ =∅,then loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r ,c robot r (X r s ,∅,∅))−c,max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P ′ x F ~ r x (t))−c) = max(max ~ r∈R ′ c ~ r ,c robot r (X r s ,∅,∅), min 0≤t<∞ max ~ r∈P ′ x F ~ r x (t))−c ≤ max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅), max ~ r∈P ′ x c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.1and X r s =X r s ) = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈P ′ x ∪{r} c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.2) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.2.3andR ′ =R\(Px ∪P x ′)). 195 Ifx r ′ c ̸=∅,then loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P ′ x ′ F ~ r x ′(t))−c,max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P ′ x F ~ r x (t))−c) = max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P ′ x ′ F ~ r x ′(t), min 0≤t<∞ max ~ r∈P ′ x F ~ r x (t))−c ≤ max(max ~ r∈R ′ c ~ r , max ~ r∈P ′ x ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈P ′ x c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.1) = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈(P ′ x ′ ∪P ′ x ) c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.2) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.2.3andR ′ =R\(Px ∪P x ′)). By definition, a legal complex-task exchange S is profitable iff loss(S) < 0, thus theprevioustheoremalsoshowsusthefollowingcorollary: Corollary6.2.1 Any legal complex-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )isprofitable. If we determine the new execution time of the complex task ^ x for each robot ~ r∈ P ′ ^ x as min 0≤t<∞ max ~ r∈P ′ ^ x F ~ r ^ x (t) for its new allocation ( X ~ r s , X ~ r c , C ~ r ) after perform- ing the complex-task exchange S, then the proof of Theorem 6.2.1 also shows that loss(S) = c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). In other words, we have thefollowingcorollary: Corollary6.2.2 Anylegalprofitablecomplex-taskexchangeS reducestheteamcostof thecurrentallocationsby|loss(S)|. Theorem6.2.2 Each profitable legal complex-task exchange is evaluated by at least onerobotintheproposal-evaluationphase. Proof: Consideranyprofitablelegalcomplex-taskexchangeS = (r,r ′ ,x,x ′ ). Ifx ′ ̸=∅, then both robotsr andr ′ evaluate the complex-task exchangeS in their task-exchange evaluation phase. Since loss(S) = c team (netloss(r,S),netloss(r ′ ,S)) < 0, we have 196 r 3 r 1 r 2 x 1 x 3 r 4 x 2 (TeamCost=8) Figure 6.7: Allocations after Complex-Task Exchange (r 1 ,r 3 ,x 3 ,x 2 ) (MiniMax Team Objective) netloss(r,S) < 0 or netloss(r ′ ,S) < 0. Thus, the complex-task exchange S is proposed by at least one robot and thus evaluated in the proposal-evaluation phase. If x ′ = ∅, the robot r must propose the complex-task exchange S to robot r ′ and thus evaluated in the proposal-evaluation phase for the following reason: First con- sider the MiniSum and MiniLat team objectives. Since robot r moves its currently allocated complex task x to robot r ′ and its currently allocated simple tasks are not changed by performing the complex-task exchange S, its robot cost does not increase by performing the complex-task exchangeS. Thus, netloss(r,S) = c robot r (X r s ,∅,∅)− c robot r (X r s ,{x},{x← t})≤ 0. Now, consider the MiniMax team objective. loss(S) = max(netloss(r,S),netloss(r ′ ,S))< 0impliesthatnetloss(r,S)< 0. . BycombiningCorollary6.2.1andTheorem6.2.2,wehave: Corollary6.2.3 Any legal complex-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) is evaluated by at least one robot in the proposal-evaluation phase. Consider again the multi-robot routing example shown in Figure 6.1 for the Mini- Maxteamobjective. Table6.3showstwocomplex-taskexchangesthatreducetheteam cost of the initial allocations, namely (r 1 ,r 2 ,x 3 ,x 2 ) with loss -2 (in which robots r 1 197 and r 2 exchange their allocated complex targets) and (r 1 ,r 3 ,x 3 ,x 2 ) with loss -4 (in which robots r 1 and r 3 exchange their allocated complex targets). Assume that the robots perform complex-task exchange (r 1 ,r 3 ,x 3 ,x 2 ). Figure 6.7 shows the resulting newallocationswithteamcost8. 6.2.2 Simple-TaskExchanges Wenowconsidersimple-taskexchangesanddevelopadistributednegotiationprocedure for constructing all legal simple-task exchanges that each reduce the team cost of the current allocations ((X r s ,X r c ,C r )) r∈R . For each robot r∈ R, let P r be the coalition of robots related to robot r that are allocated to the same complex task in the current allocations (defineP r ={r} if robotr is allocated no complex task). In the following, weexplainexactlywhichproposalstherobotssendtoeachotherandhowtheyevaluate eachother’sproposals: • Task-Announcement Phase: Each robot r ∈ R sends the other robots the information about its allocated tasks in X r s and X r c , its current robot cost c r = c robot r (X r s ,X r c ,C r )anditsreactionfunctionforitscomplextaskinX r c (ifany). • Task-ExchangeEvaluationPhase: Each robotr∈ R runs the following proce- dure: For each other robot r ′ ∈ R, it considers each legal simple-task exchange S = (r,r ′ ,x,x ′ )withx∈X r s ∪{∅}andx ′ ∈X r ′ s ∪{∅}andexecutes: – Case 1: If robotr is allocated no complex task, let X r s be the set of simple targetsallocatedtorobotrafterperformingthesimple-taskexchangeS,and R ′ =R\({r}∪P r ′)bethesetofrobotsthatarenotrelatedtorobotsr orr ′ 198 before performing the simple-task exchangeS. Then, robot r calculates its netlossofthesimple-taskexchangeS = (r,r ′ ,x,x ′ )as netloss(r,S) =      c robot r ( X r s ,∅,∅)−c r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅))−c (MiniMax), wherec =c team ~ r∈R c ~ r istheteamcostofthecurrentallocations. – Case 2: If robot r is allocated a complex task x c but robot r ′ is not allo- cated complex task x c , then robot r recomputes its reaction function for its complextaskx c as F r xc (t) =c robot r ( X r s ,{x c },x c ←t). (6.2) This re-computation is necessary since the reaction functions of robots depend on the simple tasks allocated to them and can thus change after performing simple-task exchanges. Define F ~ r xc (t) =F r x r c (t) for all robots ~ r∈ P r \{r}. Let R ′ = R\ (P r ∪P r ′). Robot r calculates its netloss of simple-taskexchangeS = (r,r ′ ,x,x ′ )as netloss(r,S) =        min 0≤t<∞ ∑ ~ r∈Pr F ~ r xc (t)− ∑ ~ r∈Pr c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t))−c (MiniMax). In both Case 1 and Case 2, robot r proposes simple-task exchange S = (r,r ′ ,x,x ′ ) to robot r ′ with value netloss(r,S) by sending ⟨S,netloss(r,S)⟩iffnetloss(r,S)< 0. 199 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (a) Thereactionfunctionofrobotr 1 afterthesimple-taskexchange (r 1 ,r 3 ,∅,x 1 ) 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (b) Thereactionfunctionofrobotr 2 afterthesimple-taskexchange (r 2 ,r 3 ,∅,x 1 ) Figure 6.8: Reaction Functions of Robots r 1 and r 2 after Simple-Task Exchanges for ExampleinFigure6.1(MiniMaxTeamObjective) 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (a) Thereactionfunctionofrobotr 3 afterthesimple-taskexchanges (r 3 ,r 1 ,x 1 ,∅) (r 3 ,r 2 ,x 1 ,∅) (r 3 ,r 4 ,x 1 ,∅) 2 4 6 8 10 12 0 14 16 18 20 2 4 6 8 10 14 12 16 18 20 visit time of x 3 robot cost (= task-completion time) (b) Thereactionfunctionofrobotr 4 afterthesimple-taskexchange (r 4 ,r 3 ,∅,x 1 ) Figure 6.9: Reaction Functions of Robots r 3 and r 4 after Simple-Task Exchanges for ExampleinFigure6.1(MiniMaxTeamObjective) – Case 3: If robots r and r ′ are allocated a common complex task x c , then robot r recomputes its reaction function for complex task x c with For- mula (6.2) and proposes simple-task exchange S = (r,r ′ ,x,x ′ ) to robot r ′ withvalue F r xc (t)bysending⟨S, F r xc (t)⟩. Consideragainthemulti-robotroutingexampleshowninFigure6.1fortheMini- Maxteamobjective. Table6.4tabulatesthesimple-taskexchangesevaluatedand proposedbyeachrobotfortheinitialallocationsshowninFigure6.7. Figures6.8 200 Robots Simple-TaskExchanges(evaluated) Simple-TaskExchanges(submitted) TaskExchanges Netloss ⟨TaskExchange,Value⟩ r 1 (r 1 ,r 3 ,∅,x 1 ) -1 ⟨(r 1 ,r 3 ,∅,x 1 ),−1⟩ r 2 (r 2 ,r 3 ,∅,x 1 ) -2 ⟨(r 2 ,r 3 ,∅,x 1 ),−2⟩ r 3 (r 3 ,r 1 ,x 1 ,∅) -2 ⟨(r 3 ,r 1 ,x 1 ,∅),−2⟩ (r 3 ,r 2 ,x 1 ,∅) -2 ⟨(r 3 ,r 2 ,x 1 ,∅),−2⟩ (r 3 ,r 4 ,x 1 ,∅) − ⟨(r 3 ,r 4 ,x 1 ,∅),Figure6.9(a)⟩ r 4 (r 4 ,r 3 ,∅,x 1 ) − ⟨(r 4 ,r 3 ,∅,x 1 ),Figure6.9(b)⟩ Table 6.4: Task-Exchange Evaluation Phase of Simple-Task Exchanges for Example in Figure6.1(MiniMaxTeamObjective) and 6.9 show the reaction functions recomputed by robots for evaluating these simple-taskexchanges. • Proposal-EvaluationPhase: Each robotr∈ R runs the following procedure: If robot r does not receive any proposal, then it does nothing. Otherwise, for each proposalofasimple-taskexchangeS = (r ′ ,r,x ′ ,x)sentfromanotherrobotr ′ ,it executes: – Case 1: If robots r and r ′ are not allocated a common complex task, then it must have evaluated the simple-task exchange S ′ = (r,r ′ ,x,x ′ ) and cal- culated the corresponding netlossnetloss(r,S ′ ) in the previous phase. The simple-task exchange S ′ is identical to the simple-task exchange S. Thus, the netlossnetloss(r,S) of robotr for the simple-task exchangeS is equal tonetloss(r,S ′ ). Robotr calculates the loss of the simple-task exchangeS as loss(S) =c team (netloss(r,S),netloss(r ′ ,S)). – Case 2: If robots r and r ′ are allocated a common complex task x c , then robot r recomputes its reaction function F r xc (t) for complex task x c with Formula (6.2). Define F ~ r xc (t) =F r xc (t) for all robots ~ r∈ P r \{r,r ′ }. Let R ′ =R\P r . Robotr calculatesthelossofthesimple-taskexchangeS as 201 Robots Simple-TaskExchanges(received) Netloss Loss Profitable ⟨TaskExchanges,Value⟩ Simple-TaskExchanges r 1 ⟨(r 3 ,r 1 ,x 1 ,∅),−2⟩ -1 -1 (r 3 ,r 1 ,x 1 ,∅) r 2 ⟨(r 3 ,r 2 ,x 1 ,∅),−2⟩ -2 -2 (r 3 ,r 2 ,x 1 ,∅) r 3 ⟨(r 1 ,r 3 ,∅,x 1 ),−1⟩ -2 -1 (r 1 ,r 3 ,∅,x 1 ) ⟨(r 2 ,r 3 ,∅,x 1 ),−2⟩ -2 -2 (r 2 ,r 3 ,∅,x 1 ) ⟨(r 4 ,r 3 ,∅,x 1 ),Figure6.9(a)⟩ − 1 r 4 ⟨(r 3 ,r 4 ,x 1 ,∅),Figure6.9(b)⟩ − 1 Table6.5: Proposal-EvaluationPhaseofSimple-TaskExchangesforExampleinFigure 6.1(MiniMaxTeamObjective) loss(S) =        min 0≤t<∞ ∑ ~ r∈Pr F ~ r xc (t)− ∑ ~ r∈Pr c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t))−c (MiniMax). Finally, we define the simple-task exchange S = (r,r ′ ,x,x ′ ) to be profitable iff loss(S)< 0. Consideragainthemulti-robotroutingexampleshowninFigure6.1fortheMini- Maxteamobjective. Table6.5tabulatestheproposalsreceivedbyeachrobotand theprofitablesimple-taskexchangesfortheinitialallocationsshowninFigure6.7. Let ^ r be robot r or r ′ in a legal simple-task exchange S = (r,r ′ ,x,x ′ ), P ^ r be the coalition of robots related to robot ^ r, ^ x be the complex task allocated to robot ^ r, R ′ = R\ (P r ∪ P r ′) be the set of robots that are not related to robots r and r ′ , and (( X ~ r s , X ~ r c , C ~ r ) ~ r∈R be the new allocations after performing the simple-task exchange S on the current allocations ((X ~ r s ,X ~ r c ,C ~ r )) ~ r∈R . Now, we show that all simple-task exchanges that each reduce the team cost of the current allocations are evaluated by at leastonerobotintheproposal-evaluationphaseofthisprocedure. Lemma6.2.4 For any legal simple-task exchange, c team ~ r∈P ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) ≥ min 0≤t<∞ c team ~ r∈P ^ r F ~ r ^ x (t)if ^ x̸=∅. 202 Proof: c team ~ r∈P ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c team ~ r∈P ^ r c robot ~ r ( X ~ r s ,{^ x},{^ x← t}) ≥ min 0≤t<∞ c team ~ r∈P ^ r c robot ~ r ( X ~ r s ,{^ x},{^ x←t}) = min 0≤t<∞ c team ~ r∈P ^ r F ~ r ^ x (t) given the definition of reaction function F ~ r ^ x (t) = c robot ~ r ( X ~ r s ,{^ x},{^ x← t}) for each robot ~ r∈P ^ r . Lemma6.2.5 For any legal simple-task exchange S, c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )foreachrobot ~ r∈R ′ . Proof: The robot cost of each robot ~ r∈ R ′ does not change after performing the simple-taskexchangeS sincethesimple-taskexchangeS doesnotchangetheallocation ofrobot ~ r: 1)thesimple-taskexchangeS doesnotre-allocateanytaskforrobot ~ r since it re-allocates only simple tasks for robots r and r ′ ; and 2) the simple-task exchange S does not change the commitment (if any) of robot ~ r since it can change only the executiontimesforthecomplextasksthatareallocatedtorobotsr orr ′ . Theorem6.2.3 Foranylegalsimple-taskexchangeS, loss(S)≤c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Proof: Consider first the MiniSum and MiniLat team objectives. If robotsr andr ′ arebothallocatednocomplextask,then loss(S) = netloss(r,S)+netloss(r ′ ,S) = c robot r ( X r s ,∅,∅)−cr +c robot r ′ ( X r ′ s ,∅,∅)−c r ′ = ∑ ~ r∈{r;r ′ } c robot ~ r ( X ~ r s , X ~ r c , C ~ r )+ ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈{r;r ′ } c ~ r − ∑ ~ r∈R ′ c ~ r (Lemma6.2.5) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c ~ r (R ′ =R\{r,r ′ }) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). 203 If robot r is allocated no complex task but robot r ′ is allocated a complex task x c , then loss(S) = netloss(r,S)+netloss(r ′ ,S) = c robot r ( X r s ,∅,∅)−cr + min 0≤t<∞ ∑ ~ r∈P r ′ F ~ r xc (t)− ∑ ~ r∈P r ′ c ~ r ≤ c robot r ( X r s ,∅,∅)−cr + ∑ ~ r∈P r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈P r ′ c ~ r (Lemma6.2.4) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c ~ r (Lemma6.2.5andR ′ =R\({r}∪P r ′)) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). If robot r is allocated a complex task x c but robot r ′ is allocated no complex task, then loss(S) = netloss(r,S)+netloss(r ′ ,S) = min 0≤t<∞ ∑ ~ r∈Pr F ~ r xc (t)− ∑ ~ r∈Pr c ~ r +c robot r ′ ( X r ′ s ,∅,∅)−c r ′ ≤ ∑ ~ r∈P r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈P r c ~ r +c robot r ′ ( X r ′ s ,∅,∅)−c r ′ (Lemma6.2.4) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c ~ r (Lemma6.2.5andR ′ =R\(Pr ∪{r ′ })) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Ifrobotr isallocatedacomplextaskx c androbotr ′ isallocatedadifferentcomplex taskx ′ c ,then loss(S) = netloss(r,S)+netloss(r ′ ,S) = min 0≤t<∞ ∑ ~ r∈Pr F ~ r xc (t)− ∑ ~ r∈Pr c ~ r + min 0≤t<∞ ∑ ~ r∈P r ′ F ~ r x ′ c (t)− ∑ ~ r∈P r ′ c ~ r ≤ ∑ ~ r∈Pr c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈Pr c ~ r + ∑ ~ r∈P r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈P r ′ c ~ r (Lemma6.2.4) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c ~ r (Lemma6.2.5andR ′ =R\(Pr ∪P r ′)) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Ifrobotsr andr ′ areallocatedacommoncomplextaskx c ,thenP r =P r ′ and loss(S) = min 0≤t<∞ ∑ ~ r∈Pr F ~ r xc (t)− ∑ ~ r∈Pr c ~ r ≤ ∑ ~ r∈Pr c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈Pr c ~ r (Lemma6.2.4) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.2.5andR ′ =R\Pr). 204 Now consider the MiniMax team objective. If robot r and r ′ are both allocated no complextask,then loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅))−c,max(max ~ r∈R ′ c ~ r ,c robot r ′ ( X r ′ s ,∅,∅))−c) = max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅),c robot r ′ ( X r ′ s ,∅,∅))−c = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ),c robot r ( X r s ,∅,∅),c robot r ′ ( X r ′ s ,∅,∅))−c (Lemma6.2.5) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (R ′ =R\{r,r ′ }) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). If robot r is allocated no complex task but robot r ′ is allocated a complex task x c , then loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅))−c,max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P r ′ F ~ r xc (t))−c) = max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅), min 0≤t<∞ max ~ r∈P r ′ F ~ r xc (t))−c ≤ max(max ~ r∈R ′ c ~ r ,c robot r ( X r s ,∅,∅), max ~ r∈P r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.4) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (Lemma6.2.5andR ′ =R\({r}∪P r ′)) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). If robot r is allocated a complex task x c but robot r ′ is allocated no complex task, then loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t))−c,max(max ~ r∈R ′ c ~ r ,c robot r ′ ( X r ′ s ,∅,∅))−c) = max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t),c robot r ′ ( X r ′ s ,∅,∅))−c ≤ max(max ~ r∈R ′ c ~ r , max ~ r∈Pr c robot ~ r ( X ~ r s , X ~ r c , C ~ r ),c robot r ′ ( X r ′ s ,∅,∅))−c (Lemma6.2.4) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (Lemma6.2.5andR ′ =R\(Pr ∪{r ′ })) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). 205 Ifrobotr isallocatedacomplextaskx c androbotr ′ isallocatedadifferentcomplex taskx ′ c ,thenwehave loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t))−c,max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈P r ′ F ~ r x ′ c (t))−c) = max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t), min 0≤t<∞ max ~ r∈P r ′ F ~ r x ′ c (t))−c ≤ max(max ~ r∈R ′ c ~ r , max ~ r∈Pr c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈P r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.4) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (Lemma6.2.5andR ′ =R\(Pr ∪P r ′)) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Ifrobotsr andr ′ areallocatedacommoncomplextaskx c ,thenP r =P r ′ and loss(S) = max(max ~ r∈R ′ c ~ r , min 0≤t<∞ max ~ r∈Pr F ~ r xc (t))−c ≤ max(max ~ r∈R ′ c ~ r , max ~ r∈P r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.2.4) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (Lemma6.2.5andR ′ =R\Pr) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). By definition, a legal simple-task exchangeS is profitable iffloss(S) < 0, thus the previoustheoremalsoshowsusthefollowingcorollary: Corollary6.2.4 Any legal simple-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )isprofitable. If we determine the new execution time of the complex task ^ x (if ^ x̸=∅) for each robot ~ r∈ P ^ r as min 0≤t<∞ max ~ r∈P ^ r F ~ r ^ x (t) foritsnewallocation ( X ~ r s , X ~ r c , C ~ r )afterper- forming complex-task exchange S, then the proof of Theorem 6.2.3 also shows that loss(S) = c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). In other words, we have thefollowingcorollary: Corollary6.2.5 Any legal profitable simple-task exchange S reduces the team cost of thecurrentallocationsby|loss(S)|. 206 Theorem6.2.4 Each profitable legal simple-task exchange is evaluated by at least one robotintheproposal-evaluationphase. Proof: Consider any legal simple-task exchange S = (r,r ′ ,x,x ′ ) with loss(S) < 0. If robots r and r ′ are not related in the current allocations, then loss(S) = c team (netloss(r,S),netloss(r ′ ,S)) < 0, which implies that netloss(r,S) < 0 or netloss(r ′ ,S) < 0. Thus, the simple-task exchangeS is proposed by at least one robot r and thus evaluated in the proposal-evaluation phase. If robotsr andr ′ are allocated a commoncomplextask,thenthesimple-taskexchangeS isproposedbybothrobotsand thusevaluatedintheproposal-evaluationphase. BycombiningCorollary6.2.4andTheorem6.2.4,wehave: Corollary6.2.6 Any legal simple-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) is evaluated by at least one robot in the proposal-evaluation phase. Consider again the multi-robot routing example shown in Figure 6.1 for the Mini- Max team objective. Table 6.5 shows two simple-task exchanges that reduce the team costofthecurrentallocationsshowninFigure6.7,namely(r 1 ,r 3 ,∅,x 1 )withloss-1and (r 2 ,r 3 ,∅,x 1 ) with loss -2. Assume that robotsr 2 andr 3 perform simple-task exchange (r 2 ,r 3 ,∅,x 1 ). Theleft partofFigure6.2showstheresultingnewallocationswithteam cost6. 6.2.3 SequentialNegotiationswithReactionFunctions In this section, we develop a new negotiation algorithm, called sequential negotia- tions with reaction functions, where robots construct and perform one complex-task or simple-task exchange in each round so that the current team cost decreases the most until the current team cost cannot be further decreased by any single complex-task or 207 simple-task exchange. Sequential negotiations with reaction functions consist of multi- pleroundstore-allocate(complexandsimple)tasksamongrobots. Duringeachround, robots construct all profitable legal complex-task and simple-task exchanges and select the most profitable one to perform. The algorithm terminates when there are no prof- itable single task exchanges for the current allocations. Let the current allocations be ((X r s ,X r c ,C r )) r∈R . Now,weexplainexactlywhichproposalsrobotssendtoeachother andhowtheyprocesstheseproposals: • Task-Announcement Phase: This phase is identical to the task-announcement phasedescribedinSubsections6.2.1and6.2.2. • Task-ExchangeEvaluationPhase: Eachrobotr∈R firstinitializesthefollow- ingsetstotheemptyset: thesetT r c ofprofitablelegalcomplex-taskexchangesand the setT r s of profitable legal simple-task exchanges. After that, robotr evaluates and proposes complex-task exchanges as described in the task-exchange evalua- tionphaseinSubsection6.2.1. Finally,robotrevaluatesandproposessimple-task exchangesasdescribedinthetask-exchangeevaluationphaseinSubsection6.2.2. • Proposal-Evaluation Phase: Each robot r ∈ R constructs the profitable complex-task exchanges as described in the proposal-evaluation phase in Sub- section 6.2.1 and adds them to T r c , and constructs the profitable simple-task exchanges as described in the proposal-evaluation phase in Subsection 6.2.2 and addsthemtoT r s . Afterthat, robotr sendsallprofitabletaskexchangesinT r c and T r s to the other robots and then adds the profitable complex-task and simple-task exchanges received from other robots toT r c andT r s , respectively. Finally, robotr determines the most profitable task exchangeS ∗ = argmin S∈T r s ∪T r c loss(S) (All robotsbreaktiesinthesameway). 208 • ScheduleUpdatePhase: IfT r s ∪T r c ̸=∅,thentherobotsperformS ∗ onthecur- rentallocations,whichreducesthecurrentteamcostby|loss(S ∗ )|andcompletes thisround;Otherwise,thealgorithmterminates. Corollaries6.2.36.2.6showthatallsinglecomplex-taskandsimple-taskexchanges that each decrease the current team cost are evaluated by at least one robot in the proposal-evaluation phase. Thus, sequential negotiations with reaction functions con- struct and perform a single complex-task or simple-task exchange in one round so that the current team cost decreases the most. In other words, sequential negotiations with reactionfunctionsachievetheperformanceprincipleofcomplex-tasknegotiations. 6.3 OverlappingCoalitions Now we study how to extend sequential negotiations with reaction functions to solving complex-taskallocationproblemswithoverlappingcoalitions. 6.3.1 Complex-TaskExchanges We first consider complex-task exchanges and develop a distributed negotiation proce- dureforconstructingalllegalcomplex-taskexchangesthateachreducetheteamcostof the current allocations ((X r s ,X r c ,C r )) r∈R . In the following, we explain exactly which proposalstherobotssendtoeachotherandhowtheyevaluateeachother’sproposals: • Task-Announcement Phase: Each robot r ∈ R sends to the other robots the information about its allocated tasks in X r s and X r c , its current robot cost c r = c robot r (X r s ,X r c ,C r ) and its generalized reaction functionF r X r c (t r ) for its complex tasksinX r c (ifany). 209 • Task-Exchange Evaluation Phase: Each robot r ∈ R considers each legal complex-task exchange S = (r,r ′ ,x,x ′ ): Let X ~ r c be the set of complex tasks allocated to each robot ~ r ∈ R after performing the complex-task exchange S, B ={ b 1 ,..., b l ′} be the bundles of related robots after performing the complex- task exchange S, and b ~ r ∈ B be the bundle of robots that are related to robot ~ r afterperformingthecomplex-taskexchangeS. Robotr executes: – Case1: If robotsr andr ′ are not related after performing the complex-task exchangeS (thatis, b r ∩ b r ′ =∅),thenletR ′ =R\( b r ∪ b r ′ )andn b r bethe numberofcomplextasksallocatedtobundle b r . Robotr firstconstructsthe generalized reaction functionF r X r c (t r ) for the complex tasks in X r c , where t r is the vector of execution times of the complex tasks in X r c . After that, it calculatesitsnetlossofcomplex-taskexchangeS = (r,r ′ ,x,x ′ )as netloss(r,S) =          min t b r ∈R n b r + ∑ ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈ b r c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r )))−c (MiniMax), wheret b r isthevectoroftheexecutiontimesofthecomplextasksallocated tobundle b r ,f ~ r (t b r )isaprojectionfunctionthatprojectsthevectort b r onto the complex tasks allocated to robot ~ r ∈ b r in the same order and c = c team ~ r∈R c ~ r is the team cost of the current allocations. Finally, robotr proposes complex-taskexchangeS = (r,r ′ ,x,x ′ )torobotr ′ withvaluenetloss(r,S) bysending⟨S,netloss(r,S)⟩torobotr ′ iffnetloss(r,S)< 0. – Case 2: If robots r and r ′ are related after performing the complex-task exchange S (that is, b r = b r ′ ), then robot r first constructs the generalized reaction functionF r X r c (t r ) and then proposes the complex task-exchange 210 S = (r,r ′ ,x,x ′ ) to robot r ′ with valueF r X r c (t r ) by sending⟨S,F r X r c (t r )⟩ torobotr ′ . • Proposal-Evaluation Phase: Each robot r∈ R runs the following procedure: If robot r does not receive any proposal, then it does nothing. Otherwise, for each proposed complex-task exchange S = (r ′ ,r,x ′ ,x) sent by robot r ′ ∈ R, it executes: – Case 1: If robots r and r ′ are not related after performing the complex- task exchange S (that is, b r ∩ b r ′ = ∅), then robot r must have evalu- ated the complex-task exchange S ′ = (r,r ′ ,x,x ′ ) and calculated the cor- responding netloss netloss(r,S ′ ) in the previous phase. The complex-task exchange S ′ is identical to the complex-task exchange S. Thus, the net- loss netloss(r,S) of robot r for the complex-task exchange S is equal to netloss(r,S ′ ). Finally,robotr calculatesthelossofcomplex-taskexchange S = (r ′ ,r,x ′ ,x)as: loss(S) =c team (netloss(r,S),netloss(r ′ ,S)). – Case 2: If robots r and r ′ are related after performing the complex-task exchangeS (thatis, b r = b r ′ ),thenitexecutes: LetR ′ =R\ b r . Then,itfirst constructsthegeneralizedreactionfunctionF r X r c (t r )andthencalculatesthe lossofcomplex-taskexchangeS = (r ′ ,r,x ′ ,x)as: loss(S) =          min t b r ∈R n b r + ∑ ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈ b r c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r )))−c (MiniMax), 211 whereF r ′ X r ′ c (t r ′ )isthereactionfunctionsubmittedbyrobotr ′ withcomplex- task exchangeS in the previous phase andF ~ r X ~ r c (t ~ r ) for each robot ~ r∈ b r \ {r,r ′ } is the reaction function sent by robot ~ r in the task-announcement phase. Wedefineanycomplex-taskexchangeS tobeprofitableiffloss(S)< 0. Let ^ r be robot r or r ′ in a legal complex-task exchange S = (r,r ′ ,x,x ′ ), b ^ r be the bundle of robots related to the robot ^ r after performing the complex-task exchange S, R ′ =R\( b r ∪ b r ′ )bethesetofrobotsthatarenotrelatedtorobotsrandr ′ afterperform- ing the complex-task exchange S, and (( X r s , X r c , C r )) r∈R be the new allocations after performingthecomplex-taskexchangeS onthecurrentallocations ((X r s ,X r c ,C r )) r∈R . Now, we show that that all complex-task exchanges that each reduce the current team cost are evaluated by at lease one robot in the proposal-evaluation phase of this proce- dure. Lemma6.3.1 For any legal complex-task exchange, c team ~ r∈ b ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) ≥ min t b ^ r ∈R n b ^ r + c team ~ r∈ b ^ r F ~ r X ~ r c (f ~ r (t b ^ r )). Proof: If b ^ r containsonlyonearobotr ′′ (thatis,eitherrobotr orr ′ ),then X r ′′ c =∅. Thus,wehave c team ~ r∈ b ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c robot r ′′ ( X r ′′ s , X r ′′ c , C r ′′ ) = c robot r ′′ (X r ′′ s ,∅,∅) = min t b ^ r ∈R n b ^ r + c team ~ r∈ b ^ r F ~ r X ~ r c (f ~ r (t b ^ r )). If b ^ r containsmultiplerelatedrobots,wehave: 212 c team ~ r∈ b ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c team ~ r∈ b ^ r c robot ~ r (X ~ r s , X ~ r c ,{ x ~ r ← t ~ r }) ≤ min t b ^ r ∈R n b ^ r + c team ~ r∈ ~ b c robot ~ r (X ~ r s , X ~ r c ,{ x ~ r ←f ~ r (t b ^ r )}) = min t b ^ r ∈R n b ^ r + c team ~ r∈ b ^ r F ~ r X ~ r c (f ~ r (t b ^ r )) Lemma6.3.2 For any legal complex task exchange, the robots that are related to (at leastoneof)robotsr orr ′ inthecurrentallocationsareexactlytherobotsrelatedto(at leastoneof)robotsr orr ′ inthenewallocations: b r ∪b r ′ = b r ∪ b r ′ . Proof: We prove the following two statements for any robot r ′′ ∈ R with r ′′ ̸= r and r ′′ ̸= r ′ , which prove the lemma: 1) if r ′′ is related to robots r or r ′ in the current allocations, then it is still related to robotsr orr ′ in the new allocations; and 2) ifr ′′ is notrelatedtorobotsr orr ′ inthecurrentallocations,thenitisstillnotrelatedtorobots r orr ′ inthenewallocations. Wenowprovethefirststatement: Ifrobotr ′′ isrelatedtorobotsrorr ′ inthecurrent allocations, thenthereexistsatleast onecomplextaskx ′′ ∈ X r ′′ c thatisrelated tosome complextaskx ′′′ ∈X r c ∪X r ′ c inthecurrentallocations. Afterperformingcomplex-task exchangeS betweenrobotsr andr ′ ,complextaskx ′′ isstillrelatedtothecomplextask x ′′′ ∈ X r c ∪ X r ′ c inthenewallocations. Thus,robotr ′′ isstillrelatedtorobotsr orrobot r ′ inthenewallocations. We now prove the second statement by contradiction: If robot r ′′ is not related to robots r or r ′ in the current allocations, then we assume that robot r ′′ is related to robots r or r ′ in the new allocations after performing the complex-task exchange 213 S = (r,r ′ ,x,x ′ ). If we perform a new complex-task exchange S ′ = (r,r ′ ,x ′ ,x) on the new allocations, robot r ′′ is still related to robots r and r ′ in the resulting alloca- tions(giventhefirststatementweprovedabove). However,theresultingallocationsare equaltothe currentallocations sincethe complex-taskexchangeS ′ simplyreversesthe complex-taskexchangeS. Thus,wehaveacontradictionsinceweassumethatrobotr ′′ isnotrelatedtorobotsr orr ′ inthecurrentallocations. Lemma6.3.3 For each legal complex-task exchange S, c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )foreachrobot ~ r∈R ′ . Proof: The robot cost of each robot ~ r∈ R ′ does not change after performing the complex-taskexchangeS sincethecomplex-taskexchangeS doesnotchangetheallo- cationofrobot ~ r: 1)thecomplex-taskexchangeS doesnotre-allocateanytaskforrobot ~ r since it re-allocates only complex tasks for robots r and r ′ ; and 2) the complex-task exchangeS does not change the commitments (if any) of robot ~ r for the following rea- son: First,wehave ~ r∈R\(b r ∪b r ′ )since ~ r∈R ′ =R\( b r ∪ b r ′ )andb r ∪b r ′ = b r ∪ b r ′ according to Lemma 6.3.2. Thus, robot ~ r is not related to robots r or r ′ in the current allocations. Let X b ~ r be the set of complex tasks allocated to bundle b ~ r in the current allocations. We have x / ∈ X b ~ r and x ′ / ∈ X b ~ r since robot ~ r / ∈ (b r ∪ b r ′ ). Since the complex-task exchange S does not change the allocations of complex tasks in X b ~ r to robots in bundle b ~ r , the commitments of robots in b ~ r for complex tasks in X b ~ r remain thesameinthenewallocations. Theorem6.3.1 For any legal complex-task exchange S, loss(S) ≤ c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). 214 Proof: Consider first the MiniSum and MiniLat team objectives. If robotsr andr ′ arenotrelatedinthenewallocations,then b r ∩ b r ′ =∅and loss(S) = netloss(r,S)+netloss(r ′ ,S) = min t b r ∈R n b r + ∑ ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈ b r c ~ r + min t b r ′ ∈R n b r ′ + ∑ ~ r∈ b r ′ F ~ r X ~ r c (f ~ r (t b r ′ ))− ∑ ~ r∈ b r ′ c ~ r ≤ ∑ ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )+ ∑ ~ r∈ b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈ b r ∪ b r ′ c ~ r (Lemma6.3.1) = ∑ ~ r∈ b r ∪ b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈ b r ∪ b r ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) = ∑ ~ r∈ b r ∪ b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈ b r ∪ b r ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) + ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.3) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (R ′ =R\( b r ∪ b r ′ )). Ifrobotsr andr ′ arerelatedinthenewallocations,then b r = b r ′ and loss(S) = min t b r ∈R n b r + ∑ ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈ b r c ~ r ≤ ∑ ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈ b r c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.1) = ∑ ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈ b r c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) + ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.3) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (R ′ =R\ b r ). Now consider the MiniMax team objective. If robot r and r ′ are not related in the newallocations,then b r ∩ b r ′ =∅and loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r )))−c, max(max ~ r∈R ′ c ~ r , min t b r ′ ∈R n b r ′ + max ~ r∈ b r ′ F ~ r X ~ r c (f ~ r (t b r ′ )))−c) = max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r )), min t b r ′ ∈R n b r ′ + max ~ r∈ b r ′ F ~ r X ~ r c (f ~ r (t b r ′ )))−c ≤ max(max ~ r∈R ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ),max ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈ b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.1) = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ),max ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈ b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.3) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (R ′ =R\( b r ∪ b r ′ )) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). 215 Ifrobotsr andr ′ arerelatedinthenewallocations,then b r = b r ′ and loss(S) = max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈ b r F ~ r X ~ r c (f ~ r (t b r )))−c ≤ max(max ~ r∈R ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ),max ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.1) = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ),max ~ r∈ b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.3) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (R ′ =R\ b r ) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). By definition, a legal complex-task exchange S is profitable iff loss(S) < 0, thus theprevioustheoremalsoshowsusthefollowingcorollary: Corollary6.3.1 Any legal complex-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )isprofitable. If we determine the new execution times of each robot ~ r ∈ b ^ r as f ~ r (t b ^ r ), where t b ^ r = min t b ^ r ∈R n b ^ r + c team ~ r∈ b ^ r F ~ r X ~ r c (f ~ r (t b ^ r )), for its new allocation ( X ~ r s , X ~ r c , C ~ r ) after per- formingthecomplex-taskexchangeS,thentheproofofTheorem6.3.1alsoshowsthat loss(S) = c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). In other words, we have thefollowingcorollary: Corollary6.3.2 Anylegalprofitablecomplex-taskexchangeS reducestheteamcostof thecurrentallocationsby|loss(S)|. Theorem6.3.2 Each profitable legal complex-task exchange is evaluated by at least onerobotintheproposal-evaluationphase. Proof: Consider any legal complex-task exchangeS = (r,r ′ ,x,x ′ ) withloss(S) < 0. If robots r and r ′ are not related in the current allocations, then we have loss(S) = c team (netloss(r,S),netloss(r ′ ,S)) < 0, which implies that netloss(r,S) < 0 or 216 netloss(r ′ ,S) < 0. Thus, the complex-task exchange S is proposed by at least one robot and thus evaluated in the proposal-evaluation phase. If robotsr andr ′ are related inthecurrentallocations,thenthecomplex-taskexchangeS isproposedbybothrobots andthusevaluatedintheproposal-evaluationphase. BycombiningCorollary6.3.1andTheorem6.3.2,wehave: Corollary6.3.3 Any legal complex-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) is evaluated by at least one robot in the proposal-evaluation phase. 6.3.2 Simple-TaskExchanges We now consider simple-task exchanges and develop a distributed negotiation proce- dure for constructing all legal simple-task exchanges that each reduce the team cost of the current allocations ((X r s ,X r c ,C r )) r∈R . In the following, we explain exactly which proposalstherobotssendtoeachotherandhowtheyevaluateeachother’sproposals: • Task-Announcement Phase: Each robot r ∈ R sends to the other robots the information about its allocated tasks in X r s and X r c , its current robot cost c r = c robot r (X r s ,X r c ,C r ) and its generalized reaction functionF r X r c (t r ) for its complex tasksinX r c (ifany). • Task-Exchange Evaluation Phase: Each robot r ∈ R considers each legal simple-task exchange S = (r,r ′ ,x,x ′ ): Let X ~ r s be the set of simple tasks allocated to each robot ~ r ∈ R after performing the simple-task exchange S, B ={b 1 ,...,b l } be the bundles of related robots in the current allocations, and b ~ r ∈ B be the bundle of robots that are related to robot ~ r ∈ R in the current allocations. Robotr executes: 217 – Case 1: If robots r and r ′ are not related in the current allocations, that is, b r ∩b r ′ =∅, then letR ′ = R\(b r ∪b r ′ ),x r be the vector of complex tasks inX r c allocatedtorobotr andn b r bethenumberofcomplextasksallocated tobundleb r . Robotr firstconstructsitsgeneralizedreactionfunctionforits currentlyallocatedcomplextasksinX r c as F r X r c (t r ) =c robot r ( X r s ,X r c ,x r ←t r ). (6.3) For all robots ~ r∈ b r \{r}, define F ~ r X ~ r c (t ~ r ) =F ~ r X ~ r c (t ~ r ) sent by robot ~ r in the task-announcement phase. After that, robot r calculates its netloss of simple-taskexchangeS = (r,r ′ ,x,x ′ )as netloss(r,S) =        min t b r ∈R n b r + ∑ ~ r∈b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈b r c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈b r F ~ r X ~ r c (f ~ r (t b r )))−c (MiniMax). Finally, robot r proposes simple-task exchange S = (r,r ′ ,x,x ′ ) to robot r ′ with value netloss(r,S) by sending⟨S,netloss(r,S)⟩ to robot r ′ iff netloss(r,S)< 0. – Case2: If robotsr andr ′ are related in the current allocations, that is,b r = b r ′ , then robot r first constructs its generalized reaction function F r X r c (t r ) for its currently allocated complex tasks in X r c with Formula 6.3 and then proposes the simple-task exchange S = (r,r ′ ,x,x ′ ) to robot r ′ with value F r X r c (t r )bysending⟨S, F r X r c (t r )⟩torobotr ′ . 218 • Proposal-EvaluationPhase: Each robotr∈ R runs the following procedure: If robot r does not receive any proposal, then it does nothing. Otherwise, for each proposedsimple-taskexchangeS = (r ′ ,r,x ′ ,x)sentbyrobotr ′ ∈R,itexecutes: – Case 1: If robotsr andr ′ are not related in the current allocations ( that is, b r ∩ b r ′ =∅), then robot r must have evaluated the simple-task exchange S ′ = (r,r ′ ,x,x ′ ) and calculated the corresponding netlossnetloss(r,S ′ ) in the previous phase. The simple-task exchangeS ′ is identical to the simple- taskexchangeS. Thus,netloss(r,S)ofrobotr forsimple-taskexchangeS is equal tonetloss(r,S ′ ). Finally, robotr calculates the loss of simple-task exchangeS = (r ′ ,r,x ′ ,x)as: loss(S) =c team (netloss(r,S),netloss(r ′ ,S)). – Case 2: If robots r and r ′ are related in the current allocations (that is, b r = b r ′ ), then let R ′ = R\ b r . Robot r first constructs its generalized reaction function F r X r c (t r ) for its currently allocated complex tasks in X r c with Formula 6.3 and then calculates the loss of simple-task exchangeS = (r ′ ,r,x ′ ,x)as: loss(S) =        min t b r ∈R n b r + ∑ ~ r∈b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈b r c ~ r (MiniSumandMiniLat) max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈b r F ~ r X ~ r c (f ~ r (t b r )))−c (MiniMax), where F r ′ X r ′ c (t r ′ ) is the reaction function submitted by robotr ′ with simple- task exchange S in the previous phase and F ~ r X ~ r c (t ~ r ) = F ~ r X ~ r c (t ~ r ) for each 219 robot ~ r∈ b r \{r,r ′ } is the reaction function sent by robot ~ r in the task- announcementphase. Wedefineanysimple-taskexchangeS tobeprofitableiffloss(S)< 0. Let ^ r be robot r or r ′ in a legal simple-task exchange S = (r,r ′ ,x,x ′ ), b ^ r be the bundle of robots related to the robot ^ r in the current allocations, R ′ = R\ (b r ∪b r ′ ) be the set of robots that are not related to robotsr andr ′ in the current allocations, and (( X r s , X r c , C r )) r∈R be the new allocations after performing the simple-task exchangeS on the current allocations ((X r s ,X r c ,C r )) r∈R . Now, we show that that all simple-task exchanges that each reduce the current team cost are evaluated by at least one robot in theproposal-evaluationphaseofthisprocedure. Lemma6.3.4 For any legal simple-task exchange, c team ~ r∈b ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) ≥ min t b ^ r ∈R n b ^ r + c team ~ r∈b ^ r F ~ r X ~ r c (f ~ r (t b ^ r )). Proof: Ifb ^ r containsonlyasinglerobotr ′′ (thatis,eitherrobotr orr ′ ),thenX r ′′ c = ∅. Thus,wehave c team ~ r∈b ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c robot r ′′ ( X r ′′ s ,X r ′′ c , C r ′′ ) = c robot r ′′ ( X r ′′ s ,∅,∅) = min t ~ b ∈R n ~ b + c team ~ r∈ ~ b F ~ r X ~ r c (f ~ r (t ~ b )). Ifb ^ r containsmultiplerelatedrobots,wehave c team ~ r∈b ^ r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c team ~ r∈b ^ r c robot ~ r ( X ~ r s ,X ~ r c ,{x ~ r ← t ~ r }) ≥ min t b ^ r ∈R n b ^ r + c team ~ r∈b ^ r c robot ~ r ( X ~ r s ,X ~ r c ,{x ~ r ←f ~ r (t b ^ r )}) = min t b ^ r ∈R n b ^ r + c team ~ r∈b ^ r F ~ r X ~ r c (f ~ r (t b ^ r )) 220 Lemma6.3.5 For any legal simple-task exchange S, c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) = c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )foreachrobot ~ r∈R ′ . Proof: The robot cost of each robot ~ r∈ R ′ does not change after performing the simple-taskexchangeS sincethesimple-taskexchangeS doesnotchangetheallocation ofrobot ~ r: 1)thesimple-taskexchangeS doesnotre-allocateanytaskforrobot ~ r since it re-allocates only simple tasks for robots r and r ′ ; and 2) the simple-task exchange S does not change the commitments (if any) of robot ~ r since it can change only the executiontimesforthecomplextasksthatareallocatedtorobotsinb r orb r ′ . Theorem6.3.3 For any legal simple-task exchange S, loss(S) ≤ c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Proof: Consider first the MiniSum and MiniLat team objectives. If robotsr andr ′ arenotrelatedinthecurrentallocations,thenb r ∩b r ′ =∅and loss(S) = netloss(r,S)+netloss(r ′ ,S) = min t b r ∈R n b r + ∑ ~ r∈b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈b r c ~ r + min t b r ′ ∈R n b r ′ + ∑ ~ r∈b r ′ F ~ r X ~ r c (f ~ r (t b r ′ ))− ∑ ~ r∈b r ′ c ~ r ≤ ∑ ~ r∈b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈b r c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) + ∑ ~ r∈b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈b r ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.4) = ∑ ~ r∈b r ∪b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈b r ∪b r ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) + ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.5) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (R ′ =R\{b r ∪b r ′ }). 221 Ifrobotsr andr ′ arerelatedinthecurrentallocations,thenb r =b r ′ and loss(S) = min t b r ∈R n b r + ∑ ~ r∈b r F ~ r X ~ r c (f ~ r (t b r ))− ∑ ~ r∈b r c ~ r ≤ ∑ ~ r∈b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈b r c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.4) = ∑ ~ r∈b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈b r c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) + ∑ ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R ′ c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (Lemma6.3.5) = ∑ ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )− ∑ ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) (R ′ =R\b r ). NowconsidertheMiniMaxteamobjective. Ifrobotrandr ′ arenotrelatedinthecurrent allocations,thenb r ∩b r ′ =∅and loss(S) = max(netloss(r,S),netloss(r ′ ,S)) = max(max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈b r F ~ r X ~ r c (f ~ r (t b r )))−c, max(max ~ r∈R ′ c ~ r , min t b r ′ ∈R n b r ′ + max ~ r∈b r ′ F ~ r X ~ r c (f ~ r (t b r ′ )))−c) = max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈b r F ~ r X ~ r c (f ~ r (t b r )), min t b r ′ ∈R n b r ′ + max ~ r∈b r ′ F ~ r X ~ r c (f ~ r (t b r ′ )))−c ≤ max(max ~ r∈R ′ c ~ r ,max ~ r∈b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈b r ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.4) = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ), max ~ r∈b r ′ ∪b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.5) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (R ′ =R\{b r ∪b r ′ }) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Ifrobotsr andr ′ arerelatedinthecurrentallocations,thenb r =b r ′ and loss(S) = max(max ~ r∈R ′ c ~ r , min t b r ∈R n b r + max ~ r∈b r F ~ r X ~ r c (f ~ r (t b r )))−c ≤ max(max ~ r∈R ′ c ~ r ,max ~ r∈b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.4) = max(max ~ r∈R ′ c robot ~ r ( X ~ r s , X ~ r c , C ~ r ),max ~ r∈b r c robot ~ r ( X ~ r s , X ~ r c , C ~ r ))−c (Lemma6.3.5) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c (R ′ =R\b r ) = max ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−max ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). By definition, a legal simple-task exchangeS is profitable iffloss(S) < 0, thus the previoustheoremalsoshowsusthefollowingcorollary: 222 Corollary6.3.4 Any legal simple-task exchange with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r )isprofitable. If we determine the execution times of each robot ~ r ∈ ~ b as f ~ r (t ~ b ), where t ~ b = min t ~ b ∈R n ~ b + c team ~ r∈ ~ b F ~ r X ~ r c (f ~ r (t ~ b )), for its new allocation ( X ~ r s , X ~ r c , C ~ r ) after performing the simple-task exchange S, then the proof of Theorem 6.3.3 also shows that loss(S)≤ c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r )−c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ). Inotherwords,wehavethefollow- ingcorollary: Corollary6.3.5 Any legal profitable simple-task exchange S reduces the team cost of thecurrentallocationsby|loss(S)|. Theorem6.3.4 Each profitable legal simple-task exchange is evaluated by at least one robotintheproposal-evaluationphase. Proof: Consider any legal simple-task exchange S = (r,r ′ ,x,x ′ ) with loss(S) < 0. If robots r and r ′ are not related in the current allocations, then loss(S) = c team (netloss(r,S),netloss(r ′ ,S)) < 0, which implies that netloss(r,S) < 0 or netloss(r ′ ,S) < 0. Thus, the simple-task exchangeS is proposed by at least one robot andthusevaluatedintheproposal-evaluationphase. Ifrobotsr andr ′ arerelatedinthe currentallocations,thenthesimple-taskexchangeS isproposedbybothrobotsandthus evaluatedintheproposal-evaluationphase. BycombiningCorollary6.3.4andTheorem6.3.4,wehave: Corollary6.3.6 Any legal simple-task exchange S with c team ~ r∈R c robot ~ r ( X ~ r s , X ~ r c , C ~ r ) < c team ~ r∈R c robot ~ r (X ~ r s ,X ~ r c ,C ~ r ) is evaluated by at least one robot in the proposal-evaluation phase. 223 6.3.3 SequentialNegotiationswithGeneralizedReactionFunctions Now we show that sequential negotiations with reaction functions introduced in Sub- section6.2.3canbeextendedtothecaseofoverlappingcoalitionsbasedongeneralized reaction functions. Thus, we call the resulting negotiation algorithm sequential negoti- ationswithgeneralizedreactionfunctions. Sequential negotiations with generalized reaction functions consist of multiple rounds to re-allocate (complex and simple) tasks among robots. During each round, robots construct all profitable legal complex-task and simple-task exchanges and select the most profitable one to perform. The algorithm terminates when there are no prof- itable single task exchanges for the current allocations. Let the current allocations be ((X r s ,X r c ,C r )) r∈R . Now we explain exactly which proposals robots send to each other andhowtheyprocesstheseproposals: • Task-Announcement Phase: This phase is identical to the task-announcement phasedescribedinSubsections6.3.1and6.3.2. • Task-ExchangeEvaluationPhase: Eachrobotr∈R firstinitializesthefollow- ingsetstotheemptyset: thesetT r c ofprofitablelegalcomplex-taskexchangesand the setT r s of profitable legal simple-task exchanges. After that, robotr evaluates and proposes complex-task exchanges as described in the task-exchange evalua- tionphaseinSubsection6.3.1. Finally,robotrevaluatesandproposessimple-task exchangesasdescribedinthetask-exchangeevaluationphaseinSubsection6.3.2. • Proposal-Evaluation Phase: Each robot r ∈ R constructs the profitable complex-taskexchangesasdescribedintheproposal-evaluationphaseinSubsec- tion6.3.1andaddsthemtoT r c ,andconstructsprofitablesimple-taskexchangesas described in the proposal-evaluation phase in Subsection 6.3.2 and adds them to T r s . Afterthat,robotrsendsallprofitabletaskexchangesinT r c andT r s totheother 224 robotsandthenaddsprofitablecomplex-taskandsimple-taskexchangesreceived fromotherrobotstoT r c andT r s ,respectively. Finally,robotr determinesthemost profitable task exchange S ∗ = argmin S∈T r s ∪T r c loss(S) (All robots break ties in thesameway). • Schedule-UpdatePhase: IfT r s ∪T r c ̸=∅,thentherobotsperformS ∗ onthecur- rentallocations,whichreducesthecurrentteamcostby|loss(S ∗ )|andcompletes thisround. Otherwise,thealgorithmterminates. Corollaries 6.3.3 and 6.3.6 show that all single complex-task and simple-task exchanges that each decrease the current team cost can be evaluated by at least one robot in the proposal-evaluation phase. Thus, sequential negotiations with generalized reactionfunctionsconstructandperformasinglecomplex-taskorsimple-taskexchange inoneroundsothatthecurrentteamcostdecreasesthemost. Inotherwords,sequential negotiations with generalized reaction functions achieve the performance principle of complex-tasknegotiations. 6.4 ExperimentalEvaluation We now evaluate the performance of sequential negotiations with reaction functions empirically for solving complex-task allocation problems in the domain of multi-robot routing. 6.4.1 ExperimentalSettings WefollowSubsection5.5.1toevaluateourtask-allocationalgorithmsforsolvingmulti- robotroutingproblemswiththefollowingdifferentcoalitiontypesinourexperiments: 225 • DisjointCoalitions: Eachrobotcanvisitatmostonecomplextarget(thatis, the complex-targetcapacityofallrobotsisone). • OverlappingCoalitions: Eachrobotcanvisitatmostthreecomplextargets(that is,thecomplex-targetcapacityofallrobotsisthree). Wecomparethefollowingcomplex-taskallocationalgorithms: • SARF (Mixed): When solving multi-robot routing problems with disjoint coali- tions, this algorithm implements sequential auctions with reaction functions, as described in Subsection 5.3.3, without any restriction. When solving multi-robot routing problems with overlapping coalitions, this algorithm implements sequen- tialauctionswithgeneralizedreactionfunctions,asdescribedinSubsection5.4.3, withoutanyrestriction. • SNRF (Complex-Only): When solving multi-robot routing problems with dis- joint coalitions, this algorithm implements sequential negotiations with reaction functions,asdescribedinSubsection6.2.3,withtherestrictionthatonlycomplex- target exchanges are constructed and performed. When solving multi-robot rout- ing problem with overlapping coalitions, this algorithm implements sequential negotiationswithgeneralizedreactionfunctions,asdescribedinSubsection6.3.3, withthesamerestriction. • SNRF(Simple-Only): When solving multi-robot routing problems with disjoint coalitions, this algorithm implements sequential negotiations with reaction func- tions,asdescribedinSubsection6.2.3,withtherestrictionthatonlysimple-target exchanges are constructed and performed. When solving multi-robot routing problemswithoverlappingcoalitions,thisalgorithmimplementssequentialnego- tiationswithgeneralizedreactionfunctions,asdescribedinSubsection6.3.3,with thesamerestriction. 226 • SNRF (Mixed): When solving multi-robot routing problems with disjoint coali- tions, this algorithm implements sequential negotiations with reaction functions, as described in Subsection 6.2.3, without any restriction. When solving multi- robot routing problems with overlapping coalitions, this algorithm implements sequential negotiations with generalized reaction functions, as described in Sub- section6.3.3,withoutanyrestriction. • SKAuctions: Thisalgorithmimplementsthegreedyauctionalgorithmdescribed inSection5.2. • Random: This algorithm implements the randomized task-allocation algorithm describedinSubsection5.5.1. We use the same experimental settings from Subsection 5.5.1 to evaluate our task- allocationalgorithmsforsolvingmulti-robotroutingproblemswiththreedifferentkinds of terrain, different robot simple-target capacities, different approaches of reaction- functioncalculations,anddifferentsimple-targetrobotcapacityconstraints. 6.4.2 ExperimentalResults We perform experiments with different settings introduced in the previous subsection to evaluate our complex-task negotiation algorithms, as summarized in Table 6.6. The detailed experimental data and explanations can be found in Appendex A.4. We make thefollowingobservations: TeamCosts • Theteamcostsofsequentialnegotiationswithreactionfunctionsaresmallerthan theinitialallocations,nomatterhowtheinitialallocationsareproduced. 227 Results Coalition Terrain Blockage Simple-Target Robots Simple Complex Reaction-Function RobotSimple-Target Initial Type Status Capacities Targets Targets Calculations CapacityConstraint Allocations TableA.80 TeamCosts Disjoint Outdoor-Like Known 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.81 ResponseTimes Disjoint Outdoor-Like Known 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.82 TeamCosts Disjoint Empty Known 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.83 ResponseTimes Disjoint Empty Known 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.84 TeamCosts Disjoint Indoor-Like Known 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.85 ResponseTimes Disjoint Indoor-Like Known 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.86 TeamCosts Disjoint Outdoor-Like Known 2-5 10 20-50 5 Approximated Tight SARF(Mixed) TableA.87 ResponseTimes Disjoint Outdoor-Like Known 2-5 10 20-50 5 Approximated Tight SARF(Mixed) TableA.88 TeamCosts Disjoint Outdoor-Like Known 2-5 10 20-50 5 Ideal Tight SARF(Mixed) TableA.89 ResponseTimes Disjoint Outdoor-Like Known 2-5 10 20-50 5 Ideal Tight SARF(Mixed) TableA.90 TeamCosts Disjoint Outdoor-Like Known 3-∞ 10 30 5 Approximated Loose SARF(Mixed) TableA.91 ResponseTimes Disjoint Outdoor-Like Known 3-∞ 10 30 5 Approximated Loose SARF(Mixed) TableA.92 TeamCosts Disjoint Outdoor-Like Known 3 2-10 6-30 1-5 Approximated Tight Random TableA.93 ResponseTimes Disjoint Outdoor-Like Known 3 2-10 6-30 1-5 Approximated Tight Random TableA.94 TeamCosts Disjoint Outdoor-Like Unknown 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.95 RepeatedExecutions Disjoint Outdoor-Like Unknown 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.96 ResponseTimes Disjoint Outdoor-Like Unknown 3 2-10 6-30 1-5 Approximated Tight SARF(Mixed) TableA.97 TeamCosts Overlapping Outdoor-Like Known 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.98 ResponseTimes Overlapping Outdoor-Like Known 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.99 TeamCosts Overlapping Empty Known 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.100 ResponseTimes Overlapping Empty Known 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.101 TeamCosts Overlapping Indoor-Like Known 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.102 ResponseTimes Overlapping Indoor-Like Known 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.103 TeamCosts Overlapping Outdoor-Like Known 2-5 10 20-50 3 Approximated Tight SARF(Mixed) TableA.104 ResponseTimes Overlapping Outdoor-Like Known 2-5 10 20-50 3 Approximated Tight SARF(Mixed) TableA.105 TeamCosts Overlapping Outdoor-Like Known 2-5 10 20-50 3 Ideal Tight SARF(Mixed) TableA.106 ResponseTimes Overlapping Outdoor-Like Known 2-5 10 20-50 3 Ideal Tight SARF(Mixed) TableA.107 TeamCosts Overlapping Outdoor-Like Known 3-∞ 10 30 3 Approximated Loose SARF(Mixed) TableA.108 ResponseTimes Overlapping Outdoor-Like Known 3-∞ 10 30 3 Approximated Loose SARF(Mixed) TableA.109 TeamCosts Overlapping Outdoor-Like Known 3 10 30 3-7 Approximated Tight SARF(Mixed) TableA.110 ResponseTimes Overlapping Outdoor-Like Known 3 10 30 3-7 Approximated Tight SARF(Mixed) TableA.111 TeamCosts Overlapping Outdoor-Like Known 3 2-10 6-30 3 Approximated Tight Random TableA.112 ResponseTimes Overlapping Outdoor-Like Known 3 2-10 6-30 3 Approximated Tight Random TableA.113 TeamCosts Overlapping Outdoor-Like Unknown 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.114 RepeatedExecutions Overlapping Outdoor-Like Unknown 3 2-10 6-30 3 Approximated Tight SARF(Mixed) TableA.115 ResponseTimes Overlapping Outdoor-Like Unknown 3 2-10 6-30 3 Approximated Tight SARF(Mixed) Table6.6: SummaryofComplex-TaskNegotiationExperiments • Theteamcostsofsequentialnegotiationswithreactionfunctionswhentheinitial allocationsareproducedbytherandomizedalgorithmarelargerthanthosewhen theinitialallocationsareproducedbysequentialauctionswithreactionfunctions. This observation inspires us to develop complex-task negotiation algorithms that canperformmorethanonetaskexchangeatatimeasfuturework. ResponseTimes • The response times of sequential negotiations with reaction functions are small when the number of complex targets is small (less than 5), no matter how the initialallocationsareproduced. • The response times of sequential negotiations with reaction functions when the initialallocationsareproducedbytherandomizedalgorithmarelargerthanthose 228 when the initial allocations are produced by sequential auctions with reaction functions. Thisobservationmeetsourexpectationsincetherearemoreprofitable task exchange when the initial allocations are produced by the randomized algo- rithm. • Theresponsetimesofsequentialnegotiationswithgeneralizedreactionfunctions increasedramaticallywiththenumberofcomplextargets. Thisobservationmeets our expectation since the runtime of determining visit times of related complex targetsisexponentialinthenumberofrelatedcomplextargets. 6.5 Summary In this chapter, we studied negotiation algorithms for solving complex-task allocation problems. We developed a new negotiation algorithm, called sequential negotiations with reaction functions, that re-allocates simple and complex tasks among robots in each round so that the team cost decreases the most. Our theoretical results show that robotsneedtoproposeonlynumericalvaluesorreactionfunctionstootherrobots. Our experimental results show that sequential negotiations with reaction functions reduces the team costs of the initial allocations with small response times (when the number of complextasksissmall). 229 Chapter7 Conclusions Auction-like and negotiation-like protocols, which are both derived from the original contract net protocol, are becoming popular for coordinating distributed robots to exe- cute multiple tasks. Researchers have developed auction and negotiation algorithms basedontheseprotocolsforallocatingtaskstorobotsinadistributedfashion. However, the majority of these algorithms are based on either single-item auctions or single-item exchanges, that usually result in highly sub-optimal solutions and cannot be applied to allocatingcomplextaskstorobots. Thisdissertationmakesthefollowingfourcontribu- tions: • In Chapter 3, we investigated how auction algorithms can allocate more than one tasktorobotsinoneroundsothattheteamcostincreasestheleast. Wedeveloped a novel data structure, called bid trees, from which each robot can construct and submit bids to the auctioneer independently. Our theoretical results showed that thenumberofbidseachrobotneedstosubmittotheauctioneerisonlyaconstant number. Weprovedthatbidtreescansuccinctlycharacterizealllocalinformation of the robots needed by the auctioneer to allocate a given number of tasks to robots so that the team cost increases the least. We then developed the winner- determination algorithm for the auctioneer to determine such allocations with a runtimethatisonlylinearinthenumberofrobotsandindependentofthenumber of tasks. Our experimental results showed that our new auction algorithm, called sequential auctions with bundles, can reduce the team costs of existing auction algorithmswhilemaintainingtheadvantageofsmallresponsetimes. 230 • In Chapter 4, we investigated how negotiation algorithms can be used to perform multipletaskexchangesamongrobotsinoneroundsothattheteamcostdecreases the most. We developed a novel data structure, called partial k-swaps, that each robot can construct and propose to the other robots independently. Our theoreti- cal results showed that robots need to propose only profitable partial k-swaps to each other, which usually make up only a small part of their local computation results. We proved that profitable partial k-swaps can succinctly characterize all local information of robots needed by them to construct a given number of task exchanges so that the team cost decreases the most. We then developed a dis- tributed algorithm for robots to iteratively construct and perform multiple task exchanges in each round until the team cost can no longer reduced by the given numberoftaskexchanges. Ourexperimentalresultsshowedthatournewnegotia- tionalgorithm,calledsequentialnegotiationswithK-swaps,canreducetheteam costsofthegiveninitialallocationswithsmallresponsetimes. • InChapter5,weinvestigatedhowauctionalgorithmscanallocateeitherasimple or complex task to robots in each round so that the team cost increases the least. We developed a novel data structure, called reaction functions, that each robot can construct and submit to the auctioneer independently. Our theoretical results showed that the number of bids each robot needs to submit to the auctioneer is a finite number. We proved that reaction functions can succinctly characterize all localinformationoftherobotsneededbytheauctioneertoallocateoneadditional complexorsimpletasktorobotssothattheteamcostincreasestheleast. Wethen developed an approximation technique for constructing the reaction functions so thatthenumberofbidseachrobotneedstosubmittotheauctioneerisonlyacon- stant number. Our experimental results showed that our new auction algorithm, called sequential auctions with reaction functions, can reduce the team costs of 231 an existing auction algorithm while maintaining the advantage of small response times. • In Chapter 4, we investigated how negotiation algorithms can be used to perform singletaskexchangesofsimpleandcomplextasksineachroundsothattheteam cost decreases the most. Our theoretical results showed that reaction functions canalsosuccinctlycharacterizealllocalinformationofrobotsneededbythemto constructcomplex-taskandsimple-taskexchangessothattheteamcostdecreases themost. Wethendevelopedadistributedalgorithmforrobotstoiterativelycon- struct and perform a single task exchange in one round until the team cost can no longer be reduced by single task exchanges. Our experimental results showed that our new negotiation algorithm, called sequential negotiations with reaction functions, can reduce the team costs of the given initial allocations significantly withsmallresponsetimes(aslongasthenumberofcomplextasksissmall). Therefore, in this dissertation, we develop auction and negotiation algorithms can achieve the myopic optimality performance principle for solving both simple-task and complex-taskallocationproblems. Forfuturework,welistthefollowingideasforbetter evaluatingouralgorithmsordevelopingnewalgorithms: • The auction algorithm developed in this dissertation for solving complex-task allocationproblems,namely,sequentialauctionswithreactionfunctions,canallo- cateonlyoneadditionaltasktorobotsineachroundsothattheteamcostincreases the least. In order to consider more synergies among tasks and thus obtain even better allocations, one may want to allocate more than one task in each round. It is future work to extend sequential auctions with reaction functions to be able to allocate more than one complex task to robots in one round so that the team cost increasestheleast. 232 • The runtime of the auctioneer in sequential auctions with reaction functions to determinetheoptimalexecutiontimesoftherelatedcomplextasksisexponential inthenumberoftherelatedcomplextasks. Itisfutureworktoimprovethecurrent winner determination algorithm described in Subsection 5.4.2 so that sequential auctionswithreactionfunctionscanscalewellwiththenumberofcomplextasks. • Thenegotiationalgorithmdevelopedinthisdissertationforsolvingcomplex-task allocation problems, namely, sequential negotiations with reaction functions, can performonlyonesingletaskexchangebetweentworobotsinoneroundsothatthe team cost decreases the most. In order to consider more synergies among tasks and thus obtain even better allocations, one may want to perform multiple task exchanges in one round. It is future work to extend sequential negotiations with reactionfunctionstobeabletoperformmultipletaskexchangesamongrobotsin oneroundsothattheteamcostdecreasesthemost. • The auction and negotiation algorithms developed in this dissertation all assume that robots can always execute their allocated tasks successfully. Most applica- tions require robots to continue to make progress toward the given team mission evenifsomerobotsfailduringtaskexecution[Par98,Kha10,PTL12]. Itisfuture work to extend our auction and negotiation algorithms to be robust to robot fail- ures. • The auction and negotiation algorithms developed in this dissertation all assume thatrobotscanperfectlycommunicatewitheachotheratalltimesnomatterhow far away they are. Most applications require robots to have the ability to deal withlimitedcommunicationrangeandimperfectcommunication[MML08,JH10, RA10]. It is future work to extend our auction and negotiation algorithms to be robusttolimitedcommunicationrangeandimperfectcommunication. 233 • We solved task-allocation problems in unknown terrain by repeatedly executing ourauctionandnegotiationalgorithmsfromscratcheachtimewhenrobotsobtain new information about the terrain. In the literature, however, there exist search algorithms, called incremental search algorithms, that re-use information from previoussearchestosolveaseriesofsimilarsearchproblemsfasterthanbyalways searching from scratch [KLLF04, KSD06, Sun13]. It is future work to apply ideas from incremental search to our auction and negotiation algorithms in order tosolvetask-allocationproblemsinunknownterrainevenmoreefficiently. • This dissertation focuses on extending the functionality of existing auction and negotiation algorithms to (re-)allocating more than one task at a time or to (re- )allocating complex tasks. 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In Proceedings of the American Con- trolConference,pages2660–2665,2008. 265 AppendixA ExperimentalDataandExplanations A.1 Simple-TaskAuctionExperiments Weperformthefollowingexperimentstoevaluateoursimple-taskauctionalgorithms: • Weevaluatetheteamcostsoftheoptimalalgorithmandothertask-allocationalgo- rithmsover15instancesofoutdoor-liketerrainwithrandomlygeneratedlocations ofrobotsandtargets. Wevarythenumberofrobotsfrom2,4,6,and8to10,and thenumberoftargetsfrom6,12,18,and24to30. Wesettherobotcapacityto3. We set the time limit of the optimal algorithm to one hour. We report both upper andlowerboundsreturnedbytheoptimalalgorithm. (Theseboundsaredifferent in case the optimal algorithm cannot solve a multi-robot routing problem within thetimelimit.) TheresultsarereportedinTableA.1. • We evaluate the performance of our task-allocation algorithms (except for the optimalalgorithm)over100instancesofoutdoor-liketerrainwithrandomlygen- eratedlocationsofrobotsandtargets. Wevarythenumberofrobotsfrom2,4,6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.2 for the team costs, and TableA.3fortheresponsetimes. • WeevaluatetheperformanceoftheSAB(k=1,2,3)algorithmswithandwithout refinement over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We vary the number of robots from 2, 4, 6, and 8 266 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacityto3. TheresultsarereportedinTableA.4. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) over 100 instances of empty terrain with randomly generated locations of robots and targets. We vary the number of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.5 for the team costs, and TableA.6fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) over 100 instances of indoor-like terrain with randomly gen- eratedlocationsofrobotsandtargets. Wevarythenumberofrobotsfrom2,4,6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.7 for the team costs, and TableA.8fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) with different robot capacities over 100 instances of outdoor- like terrain with randomly generated locations of robots and targets. We set the number of robots to 10. We vary the number of targets from 20, 30, and 40 to 50. Wevarytherobotcapacitiesfrom2,3,and4to5. Theresultsarereportedin TableA.9fortheteamcosts,andTableA.10fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) with ideal robot-cost calculations over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We setthenumberofrobotsto10. Wevarythenumberoftargetsfrom20,30,and40 267 to50. Wevarytherobotcapacitiesfrom2,3,and4to5. Theresultsarereported inTableA.11fortheteamcosts,andTableA.12fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) with loose robot-capacity constraints over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We setthenumberofrobotsto10,andthenumberoftargetsto30. Wevarytherobot capacities from 3, 4, 5, 6, and 7 to∞. The results are reported in Table A.13 for theteamcosts,andTableA.14fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) in unknown terrain over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We vary the number of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.15 fortheteamcosts,TableA.16fortherepeatedexecutions,andTableA.17forthe cumulativeresponsetimes. Wenowexaminetheseresultsinmoredetail. 268 Comparisonof Comparisonof Comparisonof Optimal Robots Targets Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed Lower- Upper- (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance Bound Bound MiniSumTeamObjective 2 6 124.33 121.40 1.40 11/15 120.53 -0.65 13/15 114.40 4.36 14/15 114.00 150.27 153.47 113.73 113.73 4 12 223.27 213.00 4.75 9/15 199.00 5.92 11/15 198.07 0.30 11/15 181.33 329.13 286.27 177.13 177.13 6 18 287.47 259.33 9.10 10/15 258.80 -0.34 9/15 252.87 1.62 9/15 222.67 463.47 379.87 218.73 218.73 8 24 345.80 321.67 6.70 13/15 309.87 3.15 10/15 306.13 0.92 9/15 272.13 652.93 520.47 259.66 266.39 10 30 400.53 361.53 9.08 13/15 353.73 2.13 9/15 347.00 1.57 8/15 294.07 794.73 582.73 264.28 287.06 MiniMaxTeamObjective 2 6 77.53 80.80 -5.01 9/15 75.13 6.04 14/15 71.60 3.99 14/15 64.40 86.73 93.20 63.33 63.33 4 12 90.87 96.40 -6.38 6/15 81.67 14.03 15/15 78.07 3.51 12/15 63.87 108.40 116.67 49.36 55.06 6 18 88.73 88.33 -1.33 9/15 78.73 7.66 12/15 76.13 1.03 9/15 53.80 107.00 118.73 17.59 50.79 8 24 89.00 88.60 -2.45 7/15 83.80 4.41 12/15 77.73 6.11 13/15 55.00 111.13 131.80 4.40 96.00 10 30 90.47 92.27 -3.14 9/15 86.07 5.79 11/15 84.87 -0.47 10/15 50.13 111.80 120.73 5.66 65.93 MiniLatTeamObjective 2 6 241.40 243.80 -0.96 8/15 236.20 2.64 15/15 227.80 3.30 15/15 225.40 285.93 271.27 224.73 224.73 4 12 411.20 404.53 2.00 9/15 377.73 6.17 15/15 378.40 -0.37 13/15 353.47 626.27 523.60 352.07 352.07 6 18 541.60 510.67 5.61 11/15 501.20 1.51 14/15 496.00 1.08 14/15 440.67 877.87 805.73 436.27 436.27 8 24 643.07 621.60 3.18 9/15 608.67 2.08 11/15 596.93 1.87 12/15 538.40 1224.53 916.27 516.67 536.92 10 30 734.07 682.33 6.96 13/15 681.53 -0.11 11/15 667.13 1.76 13/15 575.53 1501.53 1213.80 503.07 586.07 TableA.1: Simple-TaskAuctions: ComparisonofTeamCostswithOptimalAlgorithminOutdoor-LikeTerrain Comparisonof Comparisonof Comparisonof Robots Targets Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 2 6 133.86 127.77 3.42 82/100 125.68 1.08 87/100 122.14 2.33 97/100 118.56 165.62 152.80 4 12 216.48 205.81 3.47 73/100 199.10 2.54 73/100 191.41 3.25 82/100 176.42 328.88 271.01 6 18 291.14 265.40 7.74 73/100 254.96 3.46 72/100 249.39 1.77 74/100 222.59 488.96 392.88 8 24 353.88 321.78 8.56 81/100 312.96 2.44 69/100 307.91 1.36 72/100 266.73 649.51 504.90 10 30 406.31 366.68 9.08 80/100 353.56 3.34 68/100 351.04 0.35 63/100 298.14 809.97 605.30 MiniMaxTeamObjective 2 6 82.96 81.07 0.51 77/100 76.30 4.26 87/100 71.85 4.47 95/100 68.44 94.99 93.24 4 12 84.16 87.05 -7.15 55/100 82.64 3.48 80/100 73.02 9.51 90/100 57.50 104.22 104.05 6 18 91.11 86.58 1.76 67/100 83.06 1.90 75/100 75.16 8.13 87/100 53.91 109.32 115.68 8 24 88.77 84.85 1.70 62/100 84.16 -0.78 68/100 77.21 6.73 85/100 52.36 112.51 118.54 10 30 90.27 89.65 -2.15 60/100 84.49 3.23 76/100 78.66 5.51 80/100 50.72 113.76 125.06 MiniLatTeamObjective 2 6 247.88 238.92 2.50 83/100 231.62 2.44 95/100 227.80 1.43 99/100 226.12 311.60 278.28 4 12 392.33 377.61 2.83 69/100 371.91 1.27 84/100 363.37 1.98 92/100 338.53 617.63 478.09 6 18 525.48 492.38 5.92 76/100 487.30 0.87 90/100 478.92 1.55 88/100 434.10 911.40 736.28 8 24 651.15 608.67 6.32 84/100 602.95 0.88 80/100 592.99 1.54 89/100 527.49 1213.67 994.09 10 30 725.00 677.06 6.29 84/100 669.72 0.99 80/100 662.66 1.06 84/100 577.38 1518.02 1166.06 TableA.2: Simple-TaskAuctions: ComparisonofTeamCostsinOutdoor-LikeTerrain 269 Robots Targets Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.04 0.00 0.08 8 24 0.00 0.00 0.00 0.07 0.16 0.00 0.53 10 30 0.00 0.00 0.01 0.16 0.27 0.00 2.28 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.04 0.00 0.08 8 24 0.00 0.00 0.00 0.07 0.17 0.00 0.55 10 30 0.00 0.00 0.01 0.18 0.28 0.00 2.19 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.05 0.00 0.09 8 24 0.00 0.00 0.00 0.08 0.19 0.00 0.61 10 30 0.00 0.00 0.01 0.18 0.29 0.00 2.50 TableA.3: Simple-TaskAuctions: ComparisonofResponseTimesinOutdoor-LikeTerrain SAB(k = 2) SAB(k = 3) ComparisonofTeamCosts ComparisonofTeamCosts Robots Targets withoutRefinement withRefinement withoutandwithRefinement withoutRefinement withRefinement withoutandwithRefinement TeamCost ResponseTime TeamCost ResponseTime Difference(%) Dominance TeamCost ResponseTime TeamCost ResponseTime Difference(%) Dominance MiniSumTeamObjective 2 6 127.62 0.00 125.68 0.00 1.32 96/100 124.33 0.00 122.14 0.00 1.39 97/100 4 12 202.61 0.00 199.10 0.00 1.26 78/100 196.30 0.01 191.41 0.01 2.09 87/100 6 18 259.86 0.00 254.96 0.00 1.61 80/100 252.71 0.02 249.39 0.02 1.01 67/100 8 24 312.70 0.00 312.96 0.00 -0.17 71/100 311.07 0.07 307.91 0.07 0.86 76/100 10 30 357.45 0.01 353.56 0.01 0.91 70/100 350.73 0.18 351.04 0.18 -0.44 66/100 MiniMaxTeamObjective 2 6 78.74 0.00 76.30 0.00 1.97 88/100 73.36 0.00 71.85 0.00 1.43 94/100 4 12 84.29 0.00 82.64 0.00 0.68 83/100 77.48 0.01 73.02 0.01 4.53 85/100 6 18 83.23 0.00 83.06 0.00 -1.81 73/100 77.36 0.02 75.16 0.02 1.95 78/100 8 24 83.43 0.00 84.16 0.00 -2.03 71/100 79.07 0.07 77.21 0.07 1.49 79/100 10 30 83.51 0.01 84.49 0.01 -3.24 72/100 80.18 0.18 78.66 0.18 1.07 82/100 MiniLatTeamObjective 2 6 237.68 0.00 231.62 0.00 2.20 98/100 230.64 0.00 227.80 0.00 1.09 99/100 4 12 375.97 0.00 371.91 0.00 0.97 94/100 371.15 0.01 363.37 0.01 1.71 91/100 6 18 491.26 0.00 487.30 0.00 0.76 94/100 483.74 0.02 478.92 0.02 0.88 83/100 8 24 607.67 0.00 602.95 0.00 0.70 90/100 602.57 0.07 592.99 0.08 1.50 89/100 10 30 672.82 0.01 669.72 0.01 0.38 90/100 668.26 0.18 662.66 0.18 0.83 87/100 TableA.4: Simple-TaskAuctions: ComparisonofSABAlgorithmswithandwithoutRefinement 270 Comparisonof Comparisonof Comparisonof Robots Targets Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 2 6 122.41 116.55 3.49 84/100 114.45 1.53 87/100 113.61 0.23 89/100 107.61 150.66 135.89 4 12 201.50 186.23 6.46 68/100 180.83 2.13 72/100 172.81 3.98 87/100 160.27 306.27 247.57 6 18 267.91 243.51 8.14 78/100 233.85 3.38 65/100 228.89 1.72 62/100 200.35 447.64 361.65 8 24 324.34 286.13 10.86 82/100 278.29 2.08 61/100 275.32 0.73 64/100 237.23 594.45 459.60 10 30 368.47 321.28 12.14 83/100 313.50 2.12 64/100 307.56 1.68 63/100 263.32 755.36 558.15 MiniMaxTeamObjective 2 6 75.74 75.89 -1.65 69/100 71.82 4.25 85/100 66.50 5.50 95/100 62.25 86.64 84.22 4 12 79.62 77.86 0.43 67/100 75.37 1.50 79/100 67.23 8.77 87/100 52.88 99.01 98.01 6 18 80.80 79.86 -1.66 64/100 73.93 5.50 79/100 68.12 5.68 82/100 49.39 100.75 106.01 8 24 80.49 81.04 -4.97 54/100 76.61 2.89 78/100 70.96 5.14 82/100 46.66 103.07 111.02 10 30 82.46 81.29 -1.58 56/100 73.92 7.54 83/100 69.37 4.86 82/100 44.59 107.86 115.05 MiniLatTeamObjective 2 6 225.07 218.57 2.13 81/100 212.97 2.16 95/100 209.01 1.56 97/100 205.31 279.23 250.25 4 12 371.66 347.60 5.96 78/100 342.06 1.35 88/100 336.02 1.52 89/100 313.00 569.12 471.60 6 18 485.38 454.44 6.06 81/100 445.72 1.81 88/100 437.66 1.61 84/100 392.14 833.52 674.54 8 24 585.57 547.77 6.08 75/100 542.25 0.90 74/100 534.41 1.41 90/100 468.11 1115.49 873.13 10 30 655.50 602.48 7.70 85/100 595.14 1.13 74/100 588.38 1.06 90/100 512.86 1406.04 1030.06 TableA.5: Simple-TaskAuctions: ComparisonofTeamCostsinEmptyTerrain Robots Targets Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.02 0.00 0.08 8 24 0.00 0.00 0.00 0.07 0.09 0.00 0.51 10 30 0.00 0.00 0.01 0.19 0.27 0.00 2.18 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.02 0.00 0.08 8 24 0.00 0.00 0.00 0.08 0.10 0.00 0.51 10 30 0.00 0.00 0.01 0.18 0.27 0.00 2.20 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.03 0.00 0.09 8 24 0.00 0.00 0.00 0.06 0.10 0.00 0.58 10 30 0.00 0.00 0.01 0.16 0.31 0.00 2.44 TableA.6: Simple-TaskAuctions: ComparisonofResponseTimesinEmptyTerrain 271 Comparisonof Comparisonof Comparisonof Robots Targets Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 2 6 191.46 186.57 2.06 87/100 183.58 0.69 87/100 180.66 1.54 93/100 175.26 237.80 218.16 4 12 312.10 292.26 5.37 73/100 279.67 3.46 80/100 275.37 1.32 77/100 255.00 483.10 400.36 6 18 397.07 368.30 6.54 75/100 357.82 2.49 76/100 349.94 1.89 73/100 312.24 741.97 555.25 8 24 479.70 437.41 8.29 84/100 431.15 1.19 68/100 419.59 2.25 76/100 363.00 955.22 686.60 10 30 541.18 502.56 6.58 75/100 497.84 0.61 68/100 484.76 2.38 74/100 407.88 1200.16 798.25 MiniMaxTeamObjective 2 6 122.55 121.49 0.61 84/100 119.58 -0.41 81/100 110.24 6.41 92/100 103.98 139.28 136.75 4 12 130.38 126.70 0.40 65/100 120.88 3.10 76/100 111.79 6.28 85/100 90.43 155.20 161.31 6 18 122.29 122.15 -1.91 58/100 118.55 2.02 69/100 109.85 4.87 82/100 81.42 162.93 166.55 8 24 131.62 129.53 -0.24 54/100 125.25 2.21 73/100 117.50 5.13 79/100 79.85 168.14 172.47 10 30 121.17 126.15 -7.23 43/100 123.59 0.46 72/100 112.79 7.39 85/100 77.37 167.69 171.29 MiniLatTeamObjective 2 6 347.30 340.40 1.75 88/100 337.64 0.42 94/100 332.48 1.41 94/100 328.56 439.66 382.58 4 12 563.01 535.39 4.35 77/100 524.25 2.01 86/100 517.41 1.20 94/100 488.73 893.67 709.49 6 18 722.11 687.71 4.40 75/100 678.71 1.17 85/100 666.95 1.60 89/100 612.35 1379.89 1003.55 8 24 875.11 832.61 4.67 72/100 819.17 1.34 82/100 807.03 1.42 90/100 727.87 1771.73 1287.59 10 30 977.21 942.01 3.33 73/100 936.21 0.51 73/100 922.57 1.42 87/100 818.19 2251.95 1506.91 TableA.7: Simple-TaskAuctions: ComparisonofTeamCostsinOffice-LikeTerrain Robots Targets Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.03 0.00 0.07 8 24 0.00 0.00 0.00 0.06 0.11 0.00 0.47 10 30 0.01 0.00 0.01 0.15 0.29 0.00 2.06 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.02 0.00 0.07 8 24 0.00 0.00 0.00 0.08 0.09 0.00 0.45 10 30 0.00 0.00 0.01 0.19 0.27 0.00 1.97 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.00 0.02 0.03 0.00 0.08 8 24 0.00 0.00 0.00 0.08 0.10 0.00 0.53 10 30 0.00 0.00 0.01 0.19 0.32 0.00 2.27 TableA.8: Simple-TaskAuctions: ComparisonofResponseTimesinOffice-LikeTerrain 272 Comparisonof Comparisonof Comparisonof Robots Targets Capacities Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 10 20 2 309.53 290.49 5.36 74/100 279.37 3.59 75/100 276.35 0.72 65/100 243.27 608.84 448.34 10 30 3 406.31 366.68 9.08 80/100 353.56 3.34 68/100 351.04 0.35 63/100 298.14 809.97 605.30 10 40 4 475.09 413.42 12.32 88/100 412.74 -0.35 47/100 399.41 3.12 74/100 342.85 986.39 702.76 10 50 5 531.50 457.82 13.28 94/100 455.71 -0.94 47/100 449.58 2.12 64/100 382.43 1120.32 800.79 MiniMaxTeamObjective 10 20 2 74.56 72.48 -0.35 63/100 68.65 3.21 83/100 63.25 6.37 85/100 43.63 92.19 101.87 10 30 3 90.27 89.65 -2.15 60/100 84.49 3.23 76/100 78.66 5.51 80/100 50.72 113.76 125.06 10 40 4 96.96 95.29 -0.46 56/100 92.56 1.02 67/100 87.18 4.91 76/100 55.83 133.77 139.33 10 50 5 105.72 104.98 -1.60 47/100 99.88 3.89 69/100 96.10 3.09 68/100 59.15 145.36 149.67 MiniLatTeamObjective 10 20 2 432.21 413.33 3.98 77/100 408.33 1.08 87/100 403.45 1.07 92/100 362.29 865.13 636.39 10 30 3 725.00 677.06 6.29 84/100 669.72 0.99 80/100 662.66 1.06 84/100 577.38 1518.02 1166.06 10 40 4 1058.03 973.31 7.67 85/100 966.27 0.67 65/100 955.91 1.06 86/100 830.37 2282.89 1714.05 10 50 5 1417.16 1301.26 7.82 87/100 1282.14 1.34 63/100 1268.62 0.28 77/100 1110.42 3090.28 2453.98 TableA.9: Simple-TaskAuctions: ComparisonofTeamCostsforDifferentRobotCapacities Robots Targets Capacities Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 10 20 2 0.00 0.00 0.00 0.01 0.03 0.00 0.23 10 30 3 0.00 0.00 0.01 0.16 0.27 0.00 2.28 10 40 4 0.01 0.01 0.02 0.86 1.77 0.00 12.24 10 50 5 0.01 0.01 0.08 3.41 9.96 0.01 44.86 MiniMaxTeamObjective 10 20 2 0.00 0.00 0.01 0.01 0.03 0.00 0.22 10 30 3 0.00 0.00 0.01 0.18 0.28 0.00 2.19 10 40 4 0.01 0.01 0.02 1.21 1.67 0.00 12.34 10 50 5 0.01 0.01 0.08 3.98 8.46 0.01 44.67 MiniLatTeamObjective 10 20 2 0.00 0.00 0.00 0.01 0.03 0.00 0.25 10 30 3 0.00 0.00 0.01 0.18 0.29 0.00 2.50 10 40 4 0.01 0.01 0.02 1.24 1.88 0.00 13.71 10 50 5 0.01 0.01 0.08 4.05 9.40 0.01 51.54 TableA.10: Simple-TaskAuctions: ComparisonofResponseTimesforDifferentRobotCapacities 273 Comparisonof Comparisonof Comparisonof Robots Targets Capacities Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 10 20 2 309.16 290.49 5.27 74/100 279.37 3.59 75/100 276.35 0.72 65/100 243.27 608.84 n/a 10 30 3 405.94 366.68 8.98 79/100 353.56 3.34 68/100 351.04 0.35 63/100 298.14 808.89 n/a 10 40 4 474.38 413.42 12.09 88/100 413.74 -0.35 47/100 399.41 3.12 74/100 342.85 983.14 n/a 10 50 5 530.92 457.82 13.18 93/100 460.71 -0.94 47/100 449.58 2.12 64/100 382.43 1112.88 n/a MiniMaxTeamObjective 10 20 2 74.20 72.48 -0.17 63/100 68.65 3.21 83/100 63.25 6.37 85/100 43.63 92.19 n/a 10 30 3 90.15 89.65 -1.88 60/100 84.49 3.23 76/100 78.66 5.51 80/100 50.72 113.42 n/a 10 40 4 96.79 95.29 -0.66 56/100 92.56 1.02 67/100 87.18 4.91 76/100 55.83 133.30 n/a 10 50 5 105.57 104.98 -1.46 46/100 99.88 3.89 69/100 96.10 3.09 68/100 59.15 143.91 n/a MiniLatTeamObjective 10 20 2 431.57 413.33 3.87 75/100 408.33 1.08 87/100 403.45 1.07 92/100 362.29 865.13 n/a 10 30 3 725.30 677.06 6.30 83/100 669.72 0.99 80/100 662.66 1.06 84/100 577.38 1516.50 n/a 10 40 4 1055.15 973.31 7.43 81/100 966.27 0.67 65/100 955.91 1.06 86/100 830.37 2265.31 n/a 10 50 5 1412.62 1301.26 7.54 85/100 1282.14 1.34 63/100 1268.62 1.21 77/100 1110.42 3051.88 n/a TableA.11: Simple-TaskAuctions: ComparisonofTeamCostsforIdealRobot-CostCalculations Robots Targets Capacities Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 10 20 2 0.00 0.00 0.00 0.01 0.04 0.00 n/a 10 30 3 0.00 0.00 0.01 0.22 0.38 0.00 n/a 10 40 4 0.01 0.01 0.03 1.51 2.52 0.01 n/a 10 50 5 0.02 0.02 0.12 5.86 12.17 0.02 n/a MiniMaxTeamObjective 10 20 2 0.00 0.00 0.00 0.01 0.04 0.00 n/a 10 30 3 0.00 0.00 0.01 0.22 0.37 0.00 n/a 10 40 4 0.01 0.01 0.03 1.69 2.34 0.01 n/a 10 50 5 0.02 0.02 0.11 6.67 11.37 0.02 n/a MiniLatTeamObjective 10 20 2 0.00 0.00 0.00 0.01 0.04 0.00 n/a 10 30 3 0.00 0.00 0.01 0.23 0.39 0.00 n/a 10 40 4 0.01 0.01 0.03 1.86 2.57 0.01 n/a 10 50 5 0.02 0.02 0.12 6.71 11.77 0.02 n/a TableA.12: Simple-TaskAuctions: ComparisonofResponseTimesforIdealRobot-CostCalculations 274 Comparisonof Comparisonof Comparisonof Robots Targets Capacities Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 10 30 3 406.31 366.68 9.08 80/100 353.56 3.34 68/100 351.04 0.35 63/100 298.14 809.97 605.30 10 30 4 325.81 281.97 12.55 89/100 275.69 1.86 71/100 269.02 2.16 72/100 247.63 771.93 722.83 10 30 5 296.73 255.52 13.35 96/100 251.00 1.50 74/100 247.70 1.21 76/100 233.44 766.32 765.57 10 30 6 283.37 243.67 13.68 96/100 239.41 1.40 69/100 237.49 0.75 72/100 227.36 766.32 773.14 10 30 7 274.35 235.64 13.86 98/100 233.92 0.48 64/100 231.86 0.88 86/100 224.32 766.32 772.86 10 30 ∞ 302.97 225.02 25.20 100/100 225.99 -0.46 58/100 225.39 0.26 82/100 221.46 766.32 771.50 MiniMaxTeamObjective 10 30 3 90.27 89.65 -2.15 60/100 84.49 3.23 76/100 78.66 5.51 80/100 50.72 113.76 125.06 10 30 4 64.00 53.86 12.82 78/100 52.26 2.43 81/100 50.48 3.09 90/100 38.37 124.01 91.15 10 30 5 60.66 48.85 17.21 89/100 47.66 1.95 80/100 46.05 2.90 86/100 36.40 130.51 83.16 10 30 6 62.06 46.46 23.02 91/100 45.77 0.91 80/100 44.50 2.44 86/100 35.98 130.51 79.96 10 30 7 63.97 46.17 25.92 93/100 45.45 1.01 81/100 44.37 2.10 85/100 35.82 130.51 78.42 10 30 ∞ 67.77 46.15 29.76 97/100 45.45 0.98 81/100 44.37 2.10 85/100 35.82 130.51 77.45 MiniLatTeamObjective 10 30 3 725.00 677.06 6.29 84/100 669.72 0.99 80/100 662.66 1.06 84/100 577.38 1518.02 1166.06 10 30 4 646.18 570.21 11.26 96/100 567.08 0.53 76/100 564.06 0.50 89/100 516.45 1621.13 971.25 10 30 5 641.27 551.45 13.44 99/100 549.38 0.35 78/100 547.09 0.39 88/100 508.93 1669.64 920.08 10 30 6 649.39 546.37 15.30 98/100 545.08 0.22 79/100 542.29 0.49 89/100 507.62 1669.64 899.11 10 30 7 661.88 545.70 16.83 98/100 544.46 0.21 78/100 541.69 0.48 91/100 507.29 1669.64 893.53 10 30 ∞ 679.37 545.68 18.83 99/100 544.44 0.21 78/100 541.67 0.48 91/100 507.29 1669.64 892.44 TableA.13: Simple-TaskAuctions: ComparisonofTeamCostsforLooseRobot-CapacityConstraints Robots Targets Capacities Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 10 30 3 0.00 0.00 0.01 0.16 0.27 0.00 2.28 10 30 4 0.00 0.00 0.01 0.32 0.61 0.00 2.23 10 30 5 0.00 0.00 0.01 0.49 1.14 0.00 2.25 10 30 6 0.00 0.00 0.02 0.68 1.81 0.00 2.22 10 30 7 0.01 0.01 0.02 0.90 2.75 0.00 2.22 10 30 ∞ 0.02 0.02 0.06 2.16 8.69 0.00 1.94 MiniMaxTeamObjective 10 30 3 0.00 0.00 0.01 0.18 0.26 0.00 2.19 10 30 4 0.00 0.00 0.01 0.35 0.53 0.00 2.18 10 30 5 0.00 0.00 0.01 0.52 0.84 0.00 2.20 10 30 6 0.00 0.00 0.01 0.59 1.06 0.00 2.23 10 30 7 0.00 0.00 0.01 0.64 1.16 0.00 2.23 10 30 ∞ 0.01 0.01 0.02 0.79 1.51 0.00 1.92 MiniLatTeamObjective 10 30 3 0.00 0.00 0.01 0.18 0.27 0.00 2.50 10 30 4 0.00 0.00 0.01 0.35 0.57 0.00 2.49 10 30 5 0.00 0.00 0.01 0.52 0.85 0.00 2.52 10 30 6 0.00 0.00 0.01 0.60 1.05 0.00 2.50 10 30 7 0.00 0.00 0.01 0.63 1.13 0.00 2.57 10 30 ∞ 0.01 0.01 0.02 0.63 1.61 0.00 2.57 TableA.14: Simple-TaskAuctions: ComparisonofResponseTimesforLooseRobot-CapacityConstraints 275 Comparisonof Comparisonof Comparisonof Robots Targets Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 2 6 160.61 159.04 1.04 88/100 158.42 -0.64 76/100 152.93 2.47 90/100 149.51 296.03 184.37 4 12 263.50 254.99 1.87 67/100 245.24 3.26 72/100 241.50 1.25 68/100 221.38 892.72 334.14 6 18 346.32 329.44 3.89 71/100 319.44 2.66 60/100 307.24 2.90 74/100 276.21 1447.18 465.81 8 24 427.69 400.77 5.79 74/100 394.50 0.99 59/100 384.86 1.95 64/100 327.99 2147.27 600.27 10 30 478.97 454.43 4.44 69/100 440.39 2.66 66/100 436.23 0.51 55/100 370.09 2874.12 704.44 MiniMaxTeamObjective 2 6 101.96 101.72 0.23 88/100 102.71 -2.31 75/100 91.61 7.38 86/100 87.97 171.64 115.44 4 12 109.49 110.34 -3.02 60/100 104.40 2.74 67/100 98.64 3.78 70/100 78.25 284.40 132.84 6 18 116.54 113.96 0.01 66/100 108.07 3.14 70/100 98.64 6.56 76/100 76.39 333.44 138.44 8 24 114.96 112.14 -0.20 60/100 110.66 -0.97 56/100 104.55 2.93 64/100 74.60 387.54 148.16 10 30 115.55 110.54 1.81 69/100 109.69 -1.85 64/100 102.24 3.67 65/100 73.23 419.30 149.60 MiniLatTeamObjective 2 6 296.08 293.06 0.76 88/100 288.24 1.31 84/100 285.88 0.70 91/100 282.66 544.71 349.40 4 12 458.87 465.21 -1.78 66/100 461.67 0.72 65/100 450.63 1.89 78/100 424.64 1525.17 616.65 6 18 608.58 606.28 -0.25 57/100 604.74 0.06 64/100 594.94 1.45 71/100 535.15 2456.60 867.52 8 24 759.03 753.17 0.35 51/100 737.09 1.97 75/100 734.45 0.34 67/100 648.45 3770.94 1154.07 10 30 816.18 819.38 -0.83 47/100 821.38 -0.38 57/100 805.58 1.75 73/100 709.61 5055.08 1319.92 TableA.15: Simple-TaskAuctions: ComparisonofTeamCostsinUnknownTerrain Comparisonof Comparisonof Comparisonof Robots Targets Parallel SAB ParallelandSAB(k = 1) SAB SAB(k = 1)andSAB(k = 2) SAB SAB(k = 2)andSAB(k = 3) Rollout Random Distributed (k = 1) Difference(%) Dominance (k = 2) Difference(%) Dominance (k = 3) Difference(%) Dominance MiniSumTeamObjective 2 6 16.07 16.31 -1.42 82/100 16.30 -3.61 79/100 15.73 1.88 89/100 15.29 22.05 23.31 4 12 24.55 24.40 -2.21 60/100 22.97 3.57 70/100 22.84 -0.90 65/100 21.27 39.67 38.33 6 18 29.16 28.11 1.07 66/100 27.38 1.38 56/100 26.82 -0.42 65/100 24.23 50.22 48.37 8 24 33.69 31.95 3.29 63/100 31.73 -1.39 54/100 31.19 0.23 64/100 26.49 59.88 58.67 10 30 34.49 33.79 0.23 59/100 32.76 1.04 59/100 31.69 1.83 63/100 27.74 64.30 66.18 MiniMaxTeamObjective 2 6 16.07 16.12 -0.15 79/100 16.95 -9.68 68/100 15.42 4.68 83/100 15.87 22.05 23.31 4 12 24.55 24.28 -1.57 66/100 24.19 -1.61 57/100 23.54 -0.98 62/100 22.16 39.67 38.33 6 18 29.16 28.84 -0.23 58/100 28.85 -1.66 55/100 27.48 3.22 63/100 27.09 50.22 48.37 8 24 33.69 32.96 0.85 61/100 32.54 -0.69 52/100 32.21 -0.64 57/100 30.09 59.88 58.67 10 30 34.49 34.00 -0.28 61/100 34.46 -2.25 51/100 34.90 -2.77 53/100 32.37 64.30 66.18 MiniLatTeamObjective 2 6 16.25 16.09 -0.16 84/100 16.11 -1.55 82/100 15.80 0.35 89/100 15.40 21.71 23.70 4 12 23.29 24.57 -7.78 57/100 24.48 -0.35 64/100 23.40 2.09 74/100 21.94 39.95 39.29 6 18 28.19 28.46 -2.34 57/100 28.82 -2.20 59/100 28.42 0.49 65/100 24.80 49.50 50.70 8 24 33.20 33.47 -1.99 50/100 32.45 2.24 64/100 32.21 0.38 68/100 28.27 58.87 60.98 10 30 33.51 33.83 -2.25 51/100 35.01 -4.24 50/100 34.08 1.93 65/100 28.49 64.77 68.81 TableA.16: Simple-TaskAuctions: ComparisonofNumbersofRepeatedExecutionsinUnknownTerrain 276 Robots Targets Parallel SAB(k = 1) SAB(k = 2) SAB(k = 3) Rollout Random Distributed MiniSumTeamObjective 2 6 0.02 0.04 0.04 0.04 0.05 0.06 0.05 4 12 0.06 0.12 0.07 0.10 0.14 0.19 0.16 6 18 0.09 0.18 0.19 0.21 0.33 0.35 0.42 8 24 0.15 0.23 0.27 0.48 0.95 0.52 1.71 10 30 0.19 0.29 0.32 1.07 2.21 0.69 5.02 MiniMaxTeamObjective 2 6 0.02 0.03 0.04 0.04 0.05 0.06 0.05 4 12 0.07 0.08 0.10 0.13 0.15 0.19 0.14 6 18 0.13 0.16 0.20 0.24 0.37 0.36 0.39 8 24 0.21 0.23 0.34 0.59 1.08 0.52 1.56 10 30 0.26 0.29 0.40 1.10 2.71 0.68 5.39 MiniLatTeamObjective 2 6 0.02 0.03 0.04 0.04 0.05 0.06 0.06 4 12 0.07 0.09 0.10 0.13 0.16 0.19 0.16 6 18 0.13 0.16 0.22 0.23 0.32 0.35 0.44 8 24 0.19 0.24 0.31 0.48 0.90 0.51 1.77 10 30 0.26 0.28 0.38 0.93 2.03 0.69 5.92 TableA.17: Simple-TaskAuctions: ComparisonofCumulativeResponseTimesinUnknownTerrain 277 r 1 x 1 x 3 r 2 x 4 x 2 FigureA.1: Example5ofMulti-RobotRouting r 1 x 1 x 3 r 2 x 4 x 2 OptimalAllocations AllocationsofParallelAuctions (TeamCost=7) r 1 x 1 x 3 r 2 x 4 x 2 AllocationsofSAB(k =1)withRobot Capacity2 (TeamCost=14) r 1 x 1 x 3 r 2 x 4 x 2 AllocationsofSAB(k =1)withRobot Capacity3 (TeamCost=9) FigureA.2: AllocationsforExampleinFigureA.1(MiniMaxTeamObjective) Table A.1 tabulates the team costs of our task-allocation algorithms with the opti- malalgorithminoutdoor-liketerrainwithasmallsamplesize(15instances). Wemake thefollowingobservations: • Thelowerandupperboundsoftheoptimalalgorithmaredifferentforinstancesof multi-robotroutingproblemswith8(ormore)robotsand24(ormore)targets(no matterwhattheteamobjectiveis). Thisimpliesthatwecannotquicklydetermine the smallest team cost of a multi-robot routing problem when its size is large, whichmeetsourexpectationsinceitisknowntobeNP-hardtosolvemulti-robot routingproblemsoptimally. 278 BidConstruction WinnerDetermination ParallelAuctions SAB(k =1) ParallelAuctions SAB(k =1) Round1 Robotr 1 : 2ontargetx 1 Robotr 1 : 2ontargetx 1 Theauctioneerallocates Theauctioneerallocates 10ontargetx 2 10ontargetx 2 targetx 1 torobotr 1 targetx 1 torobotr 1 5ontargetx 3 5ontargetx 3 2ontargetx 4 2ontargetx 4 Robotr 2 : 6ontargetx 1 Robotr 2 : 6ontargetx 1 2ontargetx 2 2ontargetx 2 3ontargetx 3 3ontargetx 3 10ontargetx 4 10ontargetx 4 Round2 Robotr 1 : 10ontargetx 2 Robotr 1 : 10ontargetx 2 Theauctioneerallocates Theauctioneerallocates 5ontargetx 3 5ontargetx 3 targetx 2 torobotr 2 targetx 2 torobotr 2 2ontargetx 4 6ontargetsx 4 Robotr 2 : 2ontargetx 2 Robotr 2 : 2ontargetx 2 3ontargetx 3 3ontargetx 3 10ontargetx 4 10ontargetsx 4 Round3 Robotr 1 : 5ontargetx 3 Robotr 1 : 5ontargetx 3 Theauctioneerallocates Theauctioneerallocates 2ontargetx 4 6ontargetx 4 targetx 4 torobotr 1 targetx 3 torobotr 1 Robotr 2 : 3ontargetx 3 Robotr 2 : 7ontargetx 3 10ontargetx 4 14ontargetx 4 Round4 Robotr 1 : 5ontargetx 3 Robotr 1 : 9ontargetx 4 Theauctioneerallocates Theauctioneerallocates (whenrobotcapacityis3) (whenrobotcapacityis3) targetx 3 torobotr 2 targetx 4 torobotr 2 whenrobotcapacityis2; Robotr 2 : 3ontargetx 3 Robotr 2 : 14ontargetx 4 theauctionnerallocates targetx 4 torobotr 1 whenrobotcapacityis3 Table A.18: Comparison of Parallel Auctions and SAB (k = 1) for Example in Figure A.1(MiniMaxTeamObjective) • Theteamcostsoftherolloutalgorithmaresmallerthanthoseofothernon-optimal algorithms and at most 15 percent larger than those of the optimal algorithm (no matter what the team objective is). The team costs of the rollout algorithm are small because the rollout algorithm is much less greedy than other non-optimal algorithms. Therolloutalgorithmmakesallocationdecisionsbasedontheevalua- tionofcompletesolutionsinsteadofthepartialonesusedbyourSABalgorithms. • The team costs of the randomized algorithm and the distributed algorithm are larger than those of the optimal algorithm (no matter what the team objective is). This observation meets our expectation since the randomized algorithm neither take into account the initial locations of the robots nor any synergies among the 279 targets;andthedistributedalgorithmdoesnotevendividethetargetsamongmul- tiplerobots. Theteamcostsofthedistributedalgorithmaresmallerthanthoseof therandomizedalgorithmfortheMiniSumandMiniLatteamobjectivesbutlarger fortheMiniMaxteamobjective. Thisinterestingobservationimpliesthateventhe simplest approach of dividingthe targetsamong multiple robots can decrease the task-completiontime. • Theteamcostsofparallelauctionsaresmallerthanthoseoftherandomizedalgo- rithm and the distributed algorithm but much larger than those of the optimal algorithm (no matter what the team objective is). This observation meets our expectation since parallel auctions take into account the initial locations of the robotsbutnotthesynergiesamongthetargets. • The team costs of the SAB (k = 1) algorithm are smaller than those of parallel auctions for the MiniSum and MiniLat team objectives but close to those for the MiniMax team objective. In Table A.1, we capture finer-scale data on the team costs of the SAB (k = 1) algorithm and parallel auctions, namely, their percent difference (calleddifference in the table) and the frequency with which the team cost of the SAB (k = 1) algorithm is not larger than the team cost of parallel auctions (called dominance in the table). One-sided binomial tests [CB90] with normal approximation (n = 100; p = 0.5; σ 2 = 50) are performed to support the hypothesis that the team costs of the SAB (k = 1) algorithm are not larger than the team costs of parallel auctions with confidence 0.995 for the MiniSum and MiniLat team objectives (which is significant) and with confidence 0.82 for MiniMax team objective (which is not significant). Figure A.1 shows an exam- ple of multi-robot routing with simple targets. There exists only one capability. Robots r 1 and r 2 provide this capability and targets x 1 , x 2 , x 3 and x 4 require it. 280 Table A.18 tabulates the allocation processes of parallel auctions and the SAB (k = 1) algorithm, and Figure A.2 shows the resulting allocations for the MiniMax team objective. For this example, parallel auctions result in much better alloca- tions than the SAB (k = 1) algorithm for the MiniMax team objective no matter whetherrobotcapacityis2or3. • TheteamcostsoftheSAB(k =2)algorithmaresmallerthanthoseoftheSAB(k = 1) algorithm, and the team costs of the SAB (k = 3) algorithm are smaller than those of the SAB (k = 2) algorithm (no matter what the team objective is). For the MiniSum team objective, the SAB (k = 2) algorithm reduces the team costs onaverageby2.79percentcomparedtotheSAB(k =1)algorithm,andtheSAB (k = 3) algorithm reduces the team costs on average by 2.27 percent compared to the SAB (k = 2) algorithm. For the MiniMax team objective, the SAB (k = 2)algorithmreducestheteamcostsonaverageby10.14percentcomparedtothe SAB (k = 1) algorithm, and the SAB (k = 3) algorithm reduces the team costs on average by 4.44 percent compared to the SAB (k = 2) algorithm. For the MiniLat team objective, the SAB (k = 2) algorithm reduces the team costs on average by 2.89 percent compared to the SAB (k = 1) algorithm, and the SAB (k =3)algorithmreducestheteamcostsonaverageby1.74percentcomparedto the SAB (k = 2) algorithm. Statistical tests support the hypothesis that the team costs of the SAB algorithms with larger bundle sizes are not larger than those of the SAB algorithms with smaller bundle sizes with confidence 0.995 (which is significant) for all team objectives. These results justify our generalization of standardsequentialsingle-itemauctionstosequentialauctionswithlargerbundle sizes. 281 • The team costs of SAB (k = 1, 2, 3) algorithms are still larger than those of the rollout algorithm (no matter what the team objective is). For the MiniSum team objective, the team costs of the SAB (k = 3) algorithm are on average 10.72 per- cent larger than those of the rollout algorithm. For the MiniMax team objective, the team costs of the SAB (k = 3) algorithm are on average 37.11 percent larger thanthoseoftherolloutalgorithm. FortheMiniLatteamobjective,theteamcosts oftheSAB(k =3)algorithmareonaverage9.49percentlargerthanthoseofthe rolloutalgorithm. Thisobservationisnotsurprisingsincesequentialauctionswith a small bundlesize evaluate partial solutions to make allocation decisions, which is much more greedy than the rollout algorithm that evaluates complete solutions tomakeallocationdecisions. Table A.2 tabulates the teamcosts of our task-allocation algorithms (except for the optimal algorithm) in outdoor-like terrain with a larger sample size (100 instances). Theseresultssupportourpreviousstatementsabouttheteamcostsofourtask-allocation algorithmsfortheresultsshowninTableA.1. TableA.3tabulatestheresponsetimesofourtask-allocationalgorithms(exceptfor theoptimalalgorithm)inoutdoor-liketerrainwith100instances. Wemakethefollowing observations: • The response times of the distributed algorithm increase dramatically with the problemsize,muchfasterthanothertask-allocationalgorithms. Thisobservation meets our expectation since robots in the distributed algorithm have to calculate their robot costs for visiting all targets, while robots in the other task-allocation algorithms need to calculate their robot costs for visiting a number of targets that isboundedbytheirrobotcapacities. 282 SAB(k =3)withoutRefinement SAB(k =3)withRefinement Capacities Team ResponseTimes ResponseTimes Cost Bidding Bidding Winner Team Bidding Bidding Winner (1stRound) (OtherRounds) Determination Cost (1stRound) (OtherRounds) Determination MiniSumTeamObjective 3 350.73 0.17 0.01 0.01 351.04 0.17 0.01 0.01 4 404.98 0.41 0.59 0.01 399.41 0.41 0.59 0.01 5 457.27 0.82 3.73 0.01 449.58 0.82 4.08 0.01 MiniMaxTeamObjective 3 80.18 0.17 0.01 0.01 78.66 0.17 0.01 0.01 4 93.32 0.41 0.66 0.01 87.18 0.41 0.77 0.01 5 101.42 0.80 3.46 0.01 96.10 0.80 3.99 0.01 MiniLatTeamObjective 3 668.26 0.17 0.01 0.01 662.66 0.17 0.01 0.01 4 965.83 0.41 0.67 0.01 955.91 0.41 0.79 0.01 5 1284.88 0.81 3.64 0.01 1278.62 0.81 4.07 0.01 Table A.19: Comparison of Response Times for SAB (k = 3) with and without Refine- ment • Theresponsetimesoftherolloutalgorithmarelargerthanthoseoftheothertask- allocation algorithms (except for the distributed algorithm) and increase quickly with the problem size. This observation meets our expectation since the rollout algorithm has to complete a partial solution in order to evaluate a single target- allocation. • Theresponsetimesoftherandomizedalgorithm,parallelauctions,andSAB(k = 1,2)algorithmsareverysmall. Thisobservationmeetsourexpectationsinceitis simpletoallocatetargetstorobotsinthesealgorithmsandrobotsneedtocalculate onlyafewrobotcosts. • TheresponsetimesoftheSAB(k =3)algorithmarelargerthanthoseoftheother task-allocation algorithms (except for the distributed and rollout algorithms) and increase quickly with the problem size. This observation meets our expectation since robots in the SAB (k = 3) algorithm have to calculate more robot costs than those in the randomized algorithm, parallel auctions and SAB (k = 1,2) algorithms. TableA.4comparestheteamcostsandresponsetimesoftheSAB(k =2,3)algo- rithmswithandwithoutrefinement. Wemakethefollowingobservations: 283 Robots Targets Capacity Parallel SAB(k=1) SAB(k=2) SAB(k=3) Rollout Random Max (Min) Max (Min) Max (Min) Max (Min) Max (Min) Max (Min) MiniSumTeamObjective 10 30 ∞ 6.58 (0.47) 8.64 (0.14) 8.57 (0.12) 8.54 (0.13) 8.30 (0.15) 5.00 (0.00) MiniMaxTeamObjective 10 30 ∞ 6.58 (0.47) 5.52 (1.19) 5.50 (1.21) 5.48 (1.19) 5.18 (1.11) 5.00 (0.00) MiniLatTeamObjective 10 30 ∞ 6.58 (0.47) 5.49 (1.19) 5.50 (1.22) 5.49 (1.22) 5.08 (1.36) 5.00 (0.00) Table A.20: Number of Targets Allocated to Robots with Loose Robot-Capacity Con- straints • TheteamcostsoftheSAB(k =2,3)algorithmswithrefinementaresmallerthan those of SAB (k = 2, 3) algorithms without refinement (no matter what the team objective is). Statistical tests support the hypothesis that the team costs of the SABalgorithmswithrefinementarenotlargerthanthosewithoutrefinementwith confidence 0.9995 (which is significant). These results justify our introduction of acautious auctioneerto sequential auctionswith bundlesbyallocating exactly one target per round. However, the difference in team costs between the SAB (k = 2, 3) algorithms with and without refinement is small (no matter what the team objective is). The SAB (k = 2, 3) algorithms with refinement reduce the team costs of the SAB (k = 2, 3) algorithms without refinement on average by at most 2percent. • The response times of the SAB (k = 2, 3) algorithms with refinement are almost identicaltothoseoftheSAB(k =2, 3)algorithmswithoutrefinement(nomatter what the team objective is). This observation is surprising since we expect the response times of the SAB algorithms with refinement to larger than those of the SAB algorithms without refinement since the number of rounds of the SAB (k = 2,3)algorithmswithrefinementisk timeslargerthanthoseoftheSAB(k =2,3) algorithms without refinement. In order to better understand the response times oftheSABalgorithms,webreakthemdownintothefollowingthreeparts: 284 – Bidding (1st Round): The computation times that robots need to calculate their bids in the first round of the SAB algorithms, as described in Subsec- tion3.3.2. – Bidding(OtherRounds): Thecomputationtimesthatrobotsneedtocalcu- late their bids in the remaining rounds of SAB algorithms. Not every robot needs to calculate its bids in the remaining rounds of the SAB algorithms. A robot needs to calculate its bids in a round only if it was allocated one or moretargetsinthepreviousround. – Winner Determination: The computation times that the auctioneer needs to determine the winning bids in the SAB algorithms, as described in Sub- section3.3.3. and perform additional experiments to compare the response times of the SAB (k =3)algorithmwithandwithoutrefinementover100instancesofoutdoor-like terrainwithrandomlygeneratedlocationsofrobotsandtargets. Wesetthenumber ofrobotsto10,andthenumberoftargetsto30. Wevarytherobotcapacitiesfrom 3, and 4 to 5. The results are reported in Table A.19. We make the following observations: – Theresponsetimesinthebidding(1stround)partareidenticalfortheSAB (k = 3) algorithm with and without refinement (no matter what the robot capacity is). This observation meets our expectation since there is no dif- ferencebetweentheSABalgorithmwithandwithoutrefinementinthebid- constructionphaseofitsfirstrounds. – The response times in the winner-determination part are small and identical for the SAB (k = 3) algorithm with and without refinement (no matter what 285 the robot capacity is). This observation meets our expectation since Theo- rem 3.3.3 shows that the auctioneer can determine the winning bids in SAB algorithms with a runtime that is linear in the number of robots and inde- pendent of the number of targets for a given bundle size. Thus, for a small bundlesizek,theincreaseintheresponsetimesforwinnerdeterminationis negligiblefortheSABalgorithmwithandwithoutrefinement. – The response times in the bidding (other rounds) part of the SAB (k = 3) algorithm with refinement are larger than those of SAB (k = 3) algorithm without refinement when the robot capacity is larger than 3. This observa- tion meets our expectation since the SAB algorithms with refinement allo- cate exactly one target to some robot per round while the SAB algorithms withoutrefinementcanallocatemorethanonetargettoonerobotperround. The number of targets allocated to a robot in the SAB algorithms without refinementmayincreasesfasterthanthatintheSABalgorithmswithrefine- ment. Thus, robots in the SAB algorithms without refinement might have fewer chances to bid on additional three targets than in the SAB algorithms withrefinement. – Theresponsetimesinthebidding(otherrounds)partaresmallandidentical for the SAB (k = 3) algorithm with and without refinement when the robot capacityis3. Thisobservationisnotsurprisingsinceanyrobotthatneedsto calculateitsbidsintheotherroundsoftheSAB(k=3)algorithmcanvisitat mosttwoadditionaltargets. Thereasonisthateachrobotneedstocalculates its bids only after it has been allocated one or more targets in the previous round. Since robots evaluate bids on at most two additional targets in the otherroundsoftheSAB(k =3)algorithmwithandwithoutrefinement, the responsetimesinthebid-constructionphaseoftheseroundsareallsmall. 286 Tables A.5 and A.7 tabulate the team costs of our task-allocation algorithms in emptyandindoor-liketerrain,respectively. Wemakethefollowingobservations: • The team costs of our task-allocation algorithms in empty terrain are all smaller than those in outdoor-like terrain, as shown in Table A.2. This observation meets our expectation since empty terrain does not contain any blocked cells and thus therobotcostofvisitingagivenbundleoftargetsinemptyterrainissmaller. • Theteamcostsofourtask-allocationalgorithmsinindoor-liketerrainarealllarger than those in outdoor-like terrain, as shown in Table A.2. This observation meets our expectation since robots in empty terrain do not have to circumnavigate any blocked cells and robots in indoor-like terrain have to go through doors at fixed positions to visit targets, whereas they only have to circumnavigate groups of randomlyblockedcellsinoutdoor-liketerrain. • The team costs of the SAB algorithms with a larger bundle size are smaller than those of the SAB algorithms with a smaller bundle size in empty and indoor- like terrain. These results demonstrate that the SAB (k≥ 2) algorithms reduce the team costs of sequential single-item auctions effectively in different kinds of terrain. TablesA.6andA.8tabulatetheresponsetimesofourtask-allocationalgorithmsin emptyandindoor-liketerrain,respectively. Theresultsarealmostidenticaltothosein outdoor-like terrain, as shown in Table A.3, and demonstrate that SAB algorithms with asmallbundlesizecanallocatetargetstorobotsefficientlyindifferentkindsofterrain. TableA.9tabulatestheteamcosts ofourtask-allocationalgorithmsinoutdoor-like terrainfordifferentrobotcapacities. Wemakethefollowingobservations: • Theteamcostsofalltask-allocationalgorithmsincreasewithrobotcapacity. This observationmeetsourexpectationsincerobotsvisitthesamenumberoftargetsas 287 their robot capacity (tight robot-capacity constraints) and the team costs increase whenrobotshavetovisitmoretargets. • TheteamcostsofSABalgorithmswithalargerbundlesizearesmallerthanthose of SAB algorithms with a smaller bundle size no matter what the robot capacity is. These results demonstrate that SAB (k≥ 2) algorithms can reduce the team costsofsequentialsingle-itemauctionseffectivelyfordifferentrobotcapacities. Table A.10 tabulates the response times of our task-allocation algorithms in outdoor-liketerrainfordifferentrobotcapacities. Wemakethefollowingobservations: • The response times of the distributed algorithm increase dramatically when the robot capacity increases. For example, its response times are more than 30 sec- onds when the robot capacity is 5. This observation meets our expectation since robots in the distributed algorithm need to calculate their robot costs for visiting alltargetsnomatterwhattherobotcapacityis. • The response times of the rollout algorithm increase quickly when the robot capacity increases. For example, its response times are close to 10 seconds when the robot capacity is 5. This observation meets our expectation since the num- ber of robot costs that the robots in the rollout algorithm need to calculate grows quicklywiththetotalnumberoftargets. • The response times of the randomized algorithm, parallel auctions, and the SAB (k =1,2)algorithmsremainsmallwhentherobotcapacityincreases. Thisobser- vation meets our expectation since the number of robot costs that the robots in these algorithms need to calculate is small and grows only slowly with the total numberoftargets. 288 • The response times of the SAB (k = 3) algorithm increase modestly when the robotcapacityincreases. Thisobservationmeetsourexpectationsincethenumber of robot costs that the robots in SAB (k = 3) algorithm need to calculate is much larger than those in the randomized algorithm, parallel auctions and SAB (k = 1, 2)algorithmbutlessthanthoseintherolloutalgorithm. TableA.11tabulatestheteamcostsofourtask-allocationalgorithmsinoutdoor-like terrainforidealrobot-costcalculations. Wemakethefollowingobservations: • We are unable to determine the team costs of the distributed algorithm within a one-hour time limit. This observation is not surprising since robots in the dis- tributed algorithm need to calculate their robot costs for visiting all targets. The runtime of calculating an exact robot cost in multi-robot routing increases expo- nentiallywiththenumberoftargets. • Theteamcostsoftheothertask-allocationalgorithmswithidealrobot-costcalcu- lationsarenotlargerthanthosewithapproximatedrobot-costcalculationsshown inTableA.9. Thisobservationmeetsourexpectationsincetheapproximatedrobot costscalculatedbythetwo-optheuristicarenotsmallerthantheexactones. Table A.12 tabulates the response times of our task-allocation algorithms in outdoor-like terrain for ideal robot-cost calculations. We observe that the response timeswithidealrobot-costcalculationsareslightlylargerthanthosewithapproximated robot-cost calculations shown in Table A.10. This observation meets our expectation sincethatarobotneedstoevaluateallpossibleroutestodeterminetheexactrobotcosts. TableA.13tabulatestheteamcostsofourtask-allocationalgorithmsinoutdoor-like terrainforlooserobot-capacityconstraints. Wemakethefollowingobservations: • The team costs of the SAB (k = 1, 2, 3) algorithms and the rollout algorithm decrease when the robot capacity increases (no matter what the team objective 289 is). This observation meets our expectation since there are more robots eligible for targets when the robot capacity increases (for a fixed number of targets). A task-allocation algorithm can usually find better allocations when there are more alternatives available for making allocation decisions. Consider again the multi- robotroutingexampleshowninFigureA.1fortheMiniMaxteamobjective. The teamcostoftheallocationsoftheSAB(k =1)algorithmwhentherobotcapacity is3issmallerthanthatwhentherobotcapacityis2,asshowninFigureA.2. • The team costs of parallel auctions and the randomized algorithm can increase when the robot capacity increases (no matter what the team objective is). This observationcanbeexplainedasfollows: Parallelauctionssimplyallocateatarget toitsnearesteligiblerobot,andtherandomizedalgorithmjustrandomlyallocates a target to any eligible robot. Since both algorithms do not take into account any synergies among targets, they can result in large robot costs when a robot is allocated a large number of targets. When the robot capacity increases, the number of targets allocated to robots becomes more unbalanced. Thus, the team costs of both algorithms can be large for large robot capacities, especially for the MiniMax team objective where the team cost is defined to be the largest robot cost. • The difference in team costs of our task-allocation algorithms with different robot capacities is small when the robot capacity is larger than 5 for the Min- iMax and MiniLat team objectives. This observation can be explained as fol- lows: Table A.20 tabulates the number of targets allocated to robots for our task- allocation algorithms with robot capacity∞ in our experiments. It shows that the number of targets allocated to robots is more balanced for the MiniMax and MiniLat team objectives. For example, the number of targets allocated to robots 290 is less than 6 on average for all our task-allocation algorithms even though each robot can visit an arbitrary number of targets. Thus, the allocations of our task- allocation algorithms remain almost the same when the robot capacity is larger than5fortheMiniMaxandMiniLatteamobjectives. • The team costs of the SAB algorithms with a larger bundle size are smaller than those of the SAB algorithms with a smaller bundle size when the robot capacity increases. However,theirpercentagedifferencedecreaseswhentherobotcapacity increases. This observation can be explained as follows: the SAB algorithms with smaller bundle size are more greedy than the SAB algorithms with larger bundle size. When the robot capacity is smaller (the robot-capacity constraint is more tight), there are less alternatives available for task-allocation algorithms to make allocation decisions and thus the penalty of making a bad decision is larger. Asaresult,theteamcostsoftheSABalgorithmswithsmallerbundlesize decreasefasterthanthoseoftheSABalgorithmswithlargerbundlesizewhenthe robot capacity increases. Consider again the multi-robot routing example shown inFigureA.1fortheMiniMaxteamobjective. TableA.21tabulatestheallocation process of the SAB (k = 2) algorithm. The resulting team cost is 7 (no matter whether the robot capacity is 2 or 3), which is 50 percent smaller than the team costofSAB(k =1)algorithmwithrobotcapacity2(asshowninthemiddlepart of Figure A.2). However, it is only 22 percent smaller than the team cost of SAB (k =1)algorithmwithrobotcapacity3(asshownintherightpartofFigureA.2). Table A.14 tabulates the response times of our task-allocation algorithms in outdoor-like terrain for loose robot-capacity constraints. We make the following observations: 291 • The response times of the randomized algorithm, parallel auctions, and the SAB (k =1,2)algorithmsremainsmallwhentherobotcapacityincreases. Thisobser- vation meets our expectation since the number of robot costs that the robots in thesealgorithmneedtocalculateissmall. • The response times of the distributed algorithm do not change much when the robot capacity increases. This observation meets our expectation since the responsetimesofthedistributedalgorithmdependmainlyonthetotalnumberof targets. • The response times of the rollout and the SAB (k = 3) algorithms increase when the robot capacity increases. This observation meets our expectation since there are more robots eligible for targets and thus more possible allocations of targets that the rollout and the SAB (k = 3) algorithms need to evaluate per round when therobot-capacityconstraintisloose. • The response times of the rollout and the SAB (k = 1, 2, 3) algorithms increase faster for the MiniSum team objective than for the MiniMax and MiniLat team objectives when the robot capacity increases. This observation is not surprising: Table A.20 shows that the number of targets allocated to robots is more unbal- anced for the MiniSum team objective. Since the computation times of robots to calculate their robot costs increase with the number of their allocated targets, it takesmoretime forrobotstocalculate theirbidsfortheMiniSum teamobjective inourexperiments. Table A.15 tabulates the team costs of our task-allocation algorithms in unknown outdoor-liketerrain. Wemakethefollowingobservations: 292 • Theteamcostsofourtask-allocationalgorithmsinunknownterrainarealllarger thanthoseinknownterrain,asshowninTableA.2(nomatterwhattheteamobjec- tive is). This observation meets our expectation since robots need to repeatedly update the blockage status of cells and revise their allocations and routes to visit targetsinunknownterrain. • The team costs of the randomized algorithm in unknown terrain are larger than those of the other task-allocation algorithms (no matter what the team objective is). This observation meets our expectation since the randomized algorithm allo- catestargetstorobotsrandomlywhenitisexecutedrepeatedly. Arobotmayhave to visit targets that are far away from its current location even though it is very closetothepreviouslyallocatedones. • TheteamcostsoftheSABalgorithmswithalargerbundlesizetendtobesmaller than those of the SAB algorithms with a smaller bundle size in unknown terrain (no matter what the team objective is). This observation meets our expectation sincealargerbundlesizereducestheteamcostsoftheSABalgorithmsinknown terrain. However,statisticaltestssupportthehypothesisthattheteamcostsofthe SAB algorithms with a larger bundle size are not larger than those of the SAB algorithms with a smaller bundle size in unknown terrain only with confidence 0.817(whichisnotsignificant). Table A.16 tabulates the numbers of repeated executions of our task-allocation algorithmsinunknownoutdoor-liketerrain. Wemakethefollowingobservations: • The number of repeated executions of the randomized and distributed algorithms are larger than those of the other task-allocation algorithms (no matter what the teamobjectiveis). Thisobservationmeetsourexpectationsincetheteamcostsof 293 therandomizedanddistributedalgorithmsarelargeinunknownterrain. Thus,the chancethatthereareblockedcellsintheirrobotroutesbecomeslarger. • The number of repeated executions of the SAB algorithms with a larger bundle size tend to be smaller than those of the SAB algorithms with a smaller bundle size (no matter what team objective is). This observation meets our expecta- tion since the team costs of the SAB algorithms with a larger bundle size are smaller in unknown terrain. However, statistical tests support the hypothesis that the repeated executions of the SAB algorithms with a larger bundle size are not largerthanthoseoftheSABalgorithmswithasmallerbundlesizeonlywithcon- fidence0.461(whichisnotsignificant). Table A.17 tabulates the cumulative response times of our task-allocation algo- rithmsinunknownoutdoor-liketerrain. Wemakethefollowingobservations: • The cumulative response times of the distributed algorithm are larger than those of other task-allocation algorithms (no matter what the team objective is). This observationmeetsourexpectationsincethenumberofrepeatedexecutionsofthe distributed algorithm is larger than that of the other algorithms (except for the randomized algorithm), as shown in Table A.16, and the response times of the distributedalgorithmarelargerthanthoseoftherandomizedalgorithminknown terrain,asshowninTableA.3. • The cumulative response times of the SAB algorithms with a larger bundle size arelargerthanthoseoftheSABalgorithmswithasmallerbundlesize(nomatter what the team objective is). This meets our expectation since the response times of the SAB algorithms with a larger bundle size are larger than those of the SAB algorithmswithasmallerbundlesizeinknownterrain,asshowninTableA.3,and thenumberofrepeatedexecutionsoftheSABalgorithmswithalargerbundlesize 294 are only slightly smaller than that of the SAB algorithms with a smaller bundle sizeinunknownterrain,asshowninTableA.16. A.2 Simple-TaskNegotiationExperiments We perform the following experiments to evaluate our simple-task negotiation algo- rithms: • Weevaluatetheteamcostsofourtask-allocationalgorithmswiththeoptimalalgo- rithmover15instancesofoutdoor-liketerrainwithrandomlygeneratedlocations ofrobotsandtargets. TheinitialallocationsfortheSNW(K =1,2,3)algorithms are produced by the SAB (k = 1) algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. We set the time limit of the optimal algorithm to one hour. TheresultsarereportedinTableA.22. • We evaluate the performance of our task-allocation algorithms (except for the optimalalgorithm)over100instancesofoutdoor-liketerrainwithrandomlygen- eratedlocationsofrobotsandtargets. TheinitialallocationsfortheSNW(K =1, 2,3)algorithmsareproducedbytheSAB(k =1)algorithm. Wevarythenumber of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.23 fortheteamcosts,andTableA.26fortheresponsetimes. • We evaluate the amount of robot communication (the number of partial multi- swaps sent by robots) in the SNW (K = 1, 2, 3) algorithms over 100 instances of outdoor-liketerrainwithrandomlygeneratedlocationsofrobotsandtargets. The initialallocationsfortheSNW(K =1,2,3)algorithmsareproducedbytheSAB 295 (k = 1) algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. TheresultsarereportedinTableA.24fortheSNW(K =1,2)algorithms,and TableA.25fortheSNW(K =3)algorithm. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) over 100 instances of empty terrain with randomly generated locations of robots and targets. The initial allocations for the SNW (K = 1, 2, 3) algorithms are produced by the SAB (k = 1) algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.27 for theteamcosts,andTableA.28fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimalalgorithm)over100instancesofindoor-liketerrainwithrandomlygener- ated locations of robots and targets. The initial allocations for the SNW (K = 1, 2,3)algorithmsareproducedbytheSAB(k =1)algorithm. Wevarythenumber of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.29 fortheteamcosts,andTableA.30fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) with different number of robot capacity over 100 instances of outdoor-liketerrainwithrandomlygeneratedlocationsofrobotsandtargets. The initialallocationsfortheSNW(K =1,2,3)algorithmsareproducedbytheSAB (k = 1) algorithm. We set the number of robots to 10. We vary the number of targets from 20, 30, and 40 to 50. We varied robot capacities from 2, 3, and 4 to 296 5. The results are reported in Table A.31 for the team costs, and Table A.32 for theresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) with ideal robot-cost calculations over 100 instances of outdoor-liketerrainwithrandomlygeneratedlocationsofrobotsandtargets. The initialallocationsfortheSNW(K =1,2,3)algorithmsareproducedbytheSAB (k = 1) algorithm. We set the number of robots to 10. We vary the number of targets from 20, 30, and 40 to 50. We vary robot capacities from 2, 3, and 4 to 5. The results are reported in Table A.33 for the team costs, and Table A.34 for the responsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimalalgorithm)over100instancesofoutdoor-liketerrainwithrandomlygen- eratedlocationsofrobotsandtargets. TheinitialallocationsfortheSNW(K =1, 2, 3) algorithms are produced by the randomized algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10, and the number of targets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reported in Table A.35 fortheteamcosts,andTableA.36fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms (except for the optimal algorithm) in unknown terrain over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. The initial allocations fortheSNW(K =1,2,3)algorithmsareproducedbytheSAB(k =1)algorithm. Wevarythenumberofrobotsfrom2,4,6,and8to10,andthenumberoftargets from 6, 12, 18, and 24 to 30. We set the robot capacity to 3. The results are reportedinTableA.37fortheteamcosts,TableA.38fortherepeatedexecutions, andTableA.39forthecumulativeresponsetimes. 297 Wenowexaminetheseresultsinmoredetail. 298 BidConstruction WinnerDetermination Bids(calculated) Bids(submitted) Allocations(evaluated) Decision Round1 Robotr 1 : 2onbundle{x 1 } Robotr 1 : 2onbundle{x 1 } 10onbundle{x 2 } 2onbundle{x 4 } ({x 1 ,x 3 }→x 1 ): 5 5onbundle{x 3 } 5onbundle{x 1 ,x 3 } 2onbundle{x 4 } 5onbundle{x 3 } 10onbundle{x 1 ,x 2 } Theauctioneer 5onbundle{x 1 ,x 3 } (x 1 →r 1 ,x 2 →r 2 ): 2 allocatestarget 6onbundle{x 1 ,x 4 } (x 1 →r 1 ): 2 x 1 torobotr 1 10onbundle{x 2 ,x 3 } (x 2 →r 2 ): 2 (bytiebreaking) 14onbundle{x 2 ,x 4 } 9onbundle{x 3 ,x 4 } Robotr 2 : 6onbundle{x 1 } Robotr 2 : 2onbundle{x 2 } 2onbundle{x 2 } 3onbundle{x 3 } 3onbundle{x 3 } 6onbundle{x 1 ,x 3 } 10onbundle{x 4 } 6onbundle{x 1 } 10onbundle{x 1 ,x 2 } 6onbundle{x 1 ,x 3 } 10onbundle{x 1 ,x 4 } 7onbundle{x 2 ,x 3 } 14onbundle{x 2 ,x 4 } 10onbundle{x 3 ,x 4 } Round2 Robotr 1 : 10onbundle{x 2 } Robotr 1 : 5onbundle{x 3 } 5onbundle{x 3 } 6onbundle{x 4 } ({x 2 ,x 3 }→r 2 ): 7 6onbundle{x 4 } Theauctioneer Robotr 2 : 2onbundle{x 2 } Robotr 2 : 2onbundle{x 2 } (x 3 →r 1 ,x 2 →r 2 ): 5 allocatestarget 3onbundle{x 3 } 3onbundle{x 3 } (x 3 →r 1 ): 5 x 2 torobotr 2 10onbundle{x 4 } 7onbundle{x 2 ,x 3 } (x 2 →r 2 ): 2 7onbundle{x 2 ,x 3 } 14onbundle{x 2 ,x 4 } 10onbundle{x 3 ,x 4 } Round3 Robotr 1 : 5onbundle{x 3 } Robotr 1 : 5onbundle{x 3 } (x 3 →r 1 ,x 4 →r 2 ): 14 6onbundle{x 4 } 6onbundle{x 4 } (whenrobotcapacityis2or3) (whenrobotcapacityis2or3) Theauctioneer 9onbundle{x 3 ,x 4 } 9onbundle{x 3 ,x 4 } (x 4 →r 1 ,x 3 →r 2 ): 7 allocatestarget (whenrobotcapacityis3) (whenrobotcapacityis3) (x 4 →r 1 ): 6 x 4 torobotr 1 (x 3 →r 2 ): 7 Robotr 2 : 7onbundle{x 3 } Robotr 2 : 7onbundle{x 3 } 14onbundle{x 4 } 14onbundle{x 4 } (whenrobotcapacityis2or3) (whenrobotcapacityis2or3) ({x 3 ,x 4 }→r 1 ): 9 14onbundle{x 3 ,x 4 } 14onbundle{x 3 ,x 4 } (whenrobotcapacityis3) (whenrobotcapacityis3) (whenrobotcapacityis3) Round4 Robotr 1 : 9onbundle{x 3 } Robotr 1 : 9onbundle{x 3 } (whenrobotcapacityis3) (whenrobotcapacityis3) Theauctioneer (x 3 →r 2 ): 7 allocatestarget Robotr 2 : 7onbundle{x 3 } Robotr 2 : 7onbundle{x 3 } x 3 torobotr 2 Table A.21: Sequential Auctions with Bundle Size Two for Example in A.1 (MiniMax TeamObjective) 299 Comparisonof Comparisonof Comparisonof Optimal Robots Targets SAB SNW SAB(k=1)andSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB Lower- Upper- (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) Bound Bound MiniSumTeamObjective 2 6 121.40 113.93 5.08 15/15 113.73 0.25 15/15 113.73 0.00 15/15 114.00 120.53 114.40 113.73 113.73 4 12 213.00 183.67 12.77 15/15 178.87 2.45 14/15 177.20 0.82 15/15 181.33 199.00 198.07 177.13 177.13 6 18 259.33 231.00 10.25 15/15 224.00 2.78 15/15 220.53 1.54 13/15 222.67 258.80 252.87 218.73 218.73 8 24 321.67 289.07 9.65 15/15 275.53 4.53 15/15 271.13 1.48 14/15 272.13 309.87 306.13 259.66 266.39 10 30 361.53 312.67 13.42 15/15 298.13 4.35 15/15 288.33 3.00 15/15 294.07 353.73 347.00 264.28 287.60 MiniMaxTeamObjective 2 6 80.80 65.00 16.93 15/15 63.33 2.19 15/15 63.33 0.00 15/15 64.40 75.13 71.60 63.33 63.33 4 12 96.40 64.33 31.56 15/15 55.80 12.14 15/15 55.33 0.79 15/15 63.87 81.67 78.07 49.36 55.06 6 18 88.33 54.87 36.79 15/15 50.60 6.26 14/15 49.53 1.96 13/15 53.80 78.73 76.13 17.59 50.79 8 24 88.60 53.80 37.03 15/15 48.73 8.20 13/15 45.80 5.48 13/15 55.00 83.80 77.73 4.40 96.00 10 30 92.27 51.47 42.54 15/15 47.33 7.33 13/15 42.87 8.45 14/15 50.13 86.07 84.87 5.66 65.93 MiniLatTeamObjective 2 6 243.80 227.27 6.16 15/15 224.73 0.66 15/15 224.73 0.00 15/15 225.40 236.20 227.80 224.73 224.73 4 12 404.53 359.20 10.69 15/15 353.73 1.35 15/15 352.93 0.19 14/15 353.47 377.73 378.40 352.07 352.07 6 18 510.67 454.00 10.34 15/15 446.53 1.58 15/15 436.40 2.24 15/15 440.67 501.20 496.00 436.27 436.27 8 24 621.60 546.00 11.73 15/15 533.20 2.17 14/15 533.20 -0.01 13/15 538.40 608.67 596.93 516.67 536.92 10 30 682.33 580.33 14.48 15/15 571.13 1.58 14/15 565.53 0.95 15/15 575.53 681.53 667.13 503.07 586.07 TableA.22: Simple-TaskNegotiations: ComparisonofTeamCostswithOptimalAlgorithminOutdoor-LikeTerrain Comparisonof Comparisonof Comparisonof Robots Targets SAB SNW SAB(k=1)andSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 2 6 127.77 120.40 4.88 100/100 118.78 1.16 100/100 118.54 0.17 100/100 118.56 125.68 122.14 4 12 205.81 180.38 11.49 100/100 176.20 2.20 95/100 174.43 0.94 100/100 176.42 199.10 191.41 6 18 265.40 229.01 13.16 100/100 222.80 2.70 96/100 219.81 1.24 98/100 222.59 254.96 249.39 8 24 321.78 283.28 11.54 100/100 269.92 4.51 93/100 262.03 2.79 98/100 266.73 312.96 307.91 10 30 366.68 314.44 13.85 100/100 300.33 4.37 97/100 292.74 2.43 96/100 298.14 353.56 351.04 MiniMaxTeamObjective 2 6 81.07 69.38 12.24 100/100 67.50 2.42 100/100 67.50 0.00 100/100 68.44 76.30 71.85 4 12 87.05 58.54 29.86 100/100 56.08 3.43 97/100 55.21 1.42 92/100 57.50 82.64 73.02 6 18 86.58 55.49 33.77 100/100 51.53 6.25 88/100 49.72 3.12 91/100 53.91 83.06 75.16 8 24 84.85 54.13 34.36 100/100 50.25 6.13 86/100 47.17 5.49 84/100 52.36 84.16 77.21 10 30 89.65 52.22 39.87 100/100 48.04 6.81 83/100 44.89 5.88 87/100 50.72 84.49 78.66 MiniLatTeamObjective 2 6 238.92 228.92 3.64 100/100 225.74 1.15 100/100 225.74 0.00 100/100 226.12 231.62 227.80 4 12 377.61 339.77 9.36 100/100 337.69 0.56 98/100 336.55 0.31 98/100 338.53 371.91 363.37 6 18 492.38 440.32 10.14 100/100 432.84 1.60 96/100 429.98 0.61 99/100 434.10 487.30 478.92 8 24 608.67 536.05 11.50 100/100 521.19 2.61 94/100 516.41 0.86 94/100 527.49 602.95 592.99 10 30 677.06 590.16 12.45 100/100 576.96 2.12 96/100 572.34 0.79 92/100 577.38 669.72 662.66 TableA.23: Simple-TaskNegotiations: ComparisonofTeamCostsinOutdoor-LikeTerrain 300 Robots Targets SNW(K=1) SNW(K=2) BeforeIteration1 Complete BeforeIteration1 BeforeIteration2 Complete Profitable All Percentage(%) Profitable All Percentage(%) Profitable All Percentage(%) Profitable All Percentage(%) Profitable All Percentage(%) MiniSumTeamObjective 2 6 2.99 18 16.61 0.33 9 3.67 6.45 36 17.92 0.00 0 N/A 0.63 9 7.00 4 12 10.51 108 9.73 1.76 54 3.26 44.41 540 8.22 34.62 1296 2.67 9.62 702 1.37 6 18 16.18 270 5.99 2.88 135 2.13 103.71 2160 4.80 98.86 6480 1.53 20.76 3375 0.62 8 24 23.24 504 4.61 3.81 252 1.51 180.65 5544 3.26 167.39 18144 0.92 26.10 9324 0.28 10 30 26.09 810 3.22 5.25 405 1.30 239.23 11340 2.11 265.10 38880 0.68 44.99 19845 0.23 MiniMaxTeamObjective 2 6 2.34 18 13.00 0.37 9 4.11 5.47 36 15.19 0.00 0 N/A 0.97 9 10.78 4 12 6.88 108 6.37 0.84 54 1.56 33.67 540 6.24 95.58 1296 7.38 63.44 702 9.04 6 18 9.13 270 3.38 1.01 135 0.75 77.56 2160 3.59 307.00 6480 4.74 193.28 3375 5.73 8 24 11.16 504 2.21 0.99 252 0.39 125.18 5544 2.26 562.31 18144 3.10 302.83 9324 3.25 10 30 14.34 810 1.77 1.12 405 0.28 205.32 11340 1.81 1073.59 38880 2.76 624.92 19845 3.15 MiniLatTeamObjective 2 6 3.20 18 17.78 0.36 9 4.00 5.93 36 16.47 0.00 0 N/A 0.65 9 7.22 4 12 10.72 108 9.93 1.78 54 3.30 41.93 540 7.76 34.53 1296 2.66 9.83 702 1.40 6 18 17.87 270 6.62 3.39 135 2.51 101.50 2160 4.70 112.80 6480 1.74 23.30 3375 0.69 8 24 23.55 504 4.67 4.82 252 1.91 158.03 5544 2.85 211.57 18144 1.17 37.07 9324 0.40 10 30 27.11 810 3.35 6.13 405 1.51 218.27 11340 1.92 302.64 38880 0.78 49.39 19845 0.25 TableA.24: Simple-TaskNegotiations: ComparisonofAmountsofRobotCommunicationforSNW(K =1,2)Algorithms 301 Robots Targets SNW(K=3) BeforeIteration1 BeforeIteration2 BeforeIteration3 Complete Profitable All Percentage(%) Profitable All Percentage(%) Profitable All Percentage(%) Profitable All Percentage(%) MiniSumTeamObjective 2 6 7.10 38 18.68 0.00 0 N/A 0.00 0 N/A 0.68 10 6.80 4 12 76.64 876 8.75 104.45 5832 1.79 15.87 3888 0.41 17.49 2328 0.75 6 18 248.66 4890 5.09 437.70 48600 0.90 128.14 38880 0.33 45.21 17970 0.25 8 24 482.80 16184 2.98 855.43 190512 0.45 237.05 163296 0.15 59.59 68320 0.09 10 30 867.19 40590 2.14 1855.90 524880 0.35 696.74 466560 0.15 138.69 185130 0.07 MiniMaxTeamObjective 2 6 5.99 38 15.76 0.00 0 N/A 0.00 0 N/A 2.31 10 23.10 4 12 60.34 876 6.89 273.58 5832 4.69 169.93 3888 4.37 202.94 2328 8.72 6 18 191.97 4890 3.93 1250.34 48600 2.57 1502.70 38880 3.86 978.17 17970 5.44 8 24 363.54 16184 2.25 2583.63 190512 1.36 3571.08 163296 2.19 1892.07 68320 2.77 10 30 705.37 40590 1.74 6253.12 524880 1.19 10545.05 466560 2.26 5278.72 185130 2.85 MiniLatTeamObjective 2 6 6.38 38 16.79 0.00 0 N/A 0.00 0 N/A 0.68 10 6.80 4 12 67.37 876 7.69 88.99 5832 1.53 16.80 3888 0.43 15.62 2328 0.67 6 18 221.11 4890 4.52 416.69 48600 0.86 149.19 38880 0.38 51.77 17970 0.29 8 24 440.17 16184 2.72 977.36 190512 0.51 391.21 163296 0.24 105.33 68320 0.15 10 30 689.49 40590 1.70 1658.42 524880 0.32 739.28 466560 0.16 154.08 185130 0.08 TableA.25: Simple-TaskNegotiations: ComparisonofAmountsofRobotCommunicationforSNW(K =3)Algorithm Robots Targets SAB(k = 1) SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.00 0.00 0.01 6 18 0.00 0.00 0.01 0.04 0.02 0.00 0.02 8 24 0.00 0.01 0.02 0.19 0.09 0.00 0.07 10 30 0.00 0.01 0.03 0.56 0.27 0.01 0.16 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.00 0.00 0.01 6 18 0.00 0.01 0.01 0.04 0.02 0.00 0.02 8 24 0.00 0.01 0.02 0.23 0.10 0.00 0.07 10 30 0.00 0.01 0.03 0.67 0.28 0.01 0.18 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.00 0.00 0.00 0.01 6 18 0.00 0.00 0.01 0.04 0.03 0.00 0.02 8 24 0.00 0.01 0.02 0.19 0.11 0.00 0.08 10 30 0.00 0.01 0.03 0.56 0.29 0.01 0.18 TableA.26: Simple-TaskNegotiations: ComparisonofResponseTimesinOutdoor-LikeTerrain 302 Comparisonof Comparisonof Comparisonof Robots Targets SAB SNW SAB(k=1)andSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 2 6 116.55 108.96 5.63 100/100 107.57 1.23 100/100 107.44 0.13 100/100 107.61 114.45 113.61 4 12 186.23 164.49 10.79 100/100 160.62 2.17 98/100 158.86 1.01 100/100 160.27 180.83 172.81 6 18 243.51 210.08 13.27 100/100 202.69 3.41 96/100 197.56 2.39 100/100 200.35 233.85 228.89 8 24 286.13 249.88 12.25 100/100 238.54 4.34 94/100 233.70 1.93 93/100 237.23 278.29 275.32 10 30 321.28 277.25 13.28 100/100 262.88 4.94 96/100 257.56 1.95 97/100 263.32 313.50 307.56 MiniMaxTeamObjective 2 6 75.89 63.29 13.89 100/100 62.17 1.65 100/100 62.17 0.00 100/100 62.25 71.82 66.50 4 12 77.86 54.41 27.30 100/100 51.88 3.97 97/100 50.68 2.07 94/100 52.88 75.37 67.23 6 18 79.86 50.12 35.06 100/100 46.33 6.47 92/100 44.27 3.89 91/100 49.39 73.93 68.12 8 24 81.04 48.21 37.99 100/100 44.73 6.33 87/100 42.10 5.27 91/100 46.66 76.61 70.96 10 30 81.29 47.53 39.86 100/100 42.70 8.74 85/100 40.20 4.74 78/100 44.59 73.92 69.37 MiniLatTeamObjective 2 6 218.57 207.31 4.60 100/100 205.33 0.82 100/100 205.27 0.03 100/100 205.31 212.97 209.01 4 12 347.60 315.20 8.80 100/100 311.54 1.03 99/100 310.74 0.24 99/100 313.00 342.06 336.02 6 18 454.44 402.34 11.07 100/100 391.48 2.49 97/100 386.78 1.16 98/100 392.14 445.72 437.66 8 24 547.77 477.47 12.41 100/100 466.85 2.14 93/100 462.39 0.88 95/100 468.11 542.25 534.41 10 30 602.48 522.62 12.95 100/100 510.22 2.31 92/100 505.96 0.82 93/100 512.86 595.14 588.38 TableA.27: Simple-TaskNegotiations: ComparisonofTeamCostsinEmptyTerrain Robots Targets SAB(k = 1) SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.00 0.01 0.00 0.01 6 18 0.00 0.00 0.01 0.04 0.02 0.00 0.02 8 24 0.00 0.01 0.01 0.17 0.09 0.00 0.07 10 30 0.00 0.01 0.03 0.53 0.27 0.01 0.19 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.01 0.01 0.05 0.02 0.00 0.02 8 24 0.00 0.01 0.02 0.22 0.10 0.00 0.08 10 30 0.00 0.01 0.03 0.67 0.27 0.01 0.18 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.00 0.01 0.00 0.01 6 18 0.00 0.00 0.01 0.04 0.03 0.00 0.02 8 24 0.00 0.01 0.02 0.17 0.10 0.00 0.06 10 30 0.00 0.01 0.03 0.54 0.31 0.01 0.16 TableA.28: Simple-TaskNegotiations: ComparisonofResponseTimesinEmptyTerrain 303 Comparisonof Comparisonof Comparisonof Robots Targets SAB SNW SAB(k=1)andSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 2 6 186.57 176.71 4.59 100/100 175.14 0.79 100/100 175.14 0.00 100/100 175.26 183.58 180.66 4 12 292.26 261.59 9.80 100/100 255.88 2.02 97/100 253.69 0.83 98/100 255.00 279.67 275.37 6 18 368.30 322.23 11.89 100/100 314.21 2.32 96/100 309.44 1.47 94/100 312.24 357.82 349.94 8 24 437.41 377.05 13.40 100/100 365.18 2.99 97/100 356.53 2.25 97/100 363.00 431.15 419.59 10 30 502.56 427.13 14.52 100/100 409.26 4.00 91/100 399.56 2.25 95/100 407.88 497.84 484.76 MiniMaxTeamObjective 2 6 121.49 103.95 11.99 100/100 102.79 0.99 100/100 102.79 0.00 100/100 103.98 119.58 110.24 4 12 126.70 89.47 27.23 100/100 84.24 5.29 95/100 83.24 1.04 93/100 90.43 120.88 111.79 6 18 122.15 80.11 32.43 100/100 76.84 3.51 86/100 74.76 2.41 89/100 81.42 118.55 109.85 8 24 129.53 82.69 34.78 100/100 73.85 9.07 84/100 69.78 4.73 82/100 79.85 125.25 117.50 10 30 126.15 77.78 36.97 100/100 70.87 7.10 79/100 66.68 4.81 82/100 77.37 123.59 112.79 MiniLatTeamObjective 2 6 340.40 329.10 2.91 100/100 328.42 0.16 100/100 328.42 0.00 100/100 328.56 337.64 332.48 4 12 535.39 494.51 7.20 100/100 488.85 1.14 100/100 486.49 0.49 98/100 488.73 524.25 517.41 6 18 687.71 614.87 10.21 100/100 610.93 0.58 96/100 607.11 0.60 98/100 612.35 678.71 666.95 8 24 832.61 738.79 10.94 100/100 722.95 2.12 93/100 715.93 0.85 97/100 727.87 819.17 807.03 10 30 942.01 825.67 12.05 100/100 810.47 1.75 96/100 804.59 0.70 91/100 818.19 936.21 922.57 TableA.29: Simple-TaskNegotiations: ComparisonofTeamCostsinIndoor-LikeTerrain Robots Targets SAB(k = 1) SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.00 0.01 0.04 0.03 0.00 0.02 8 24 0.00 0.01 0.01 0.18 0.11 0.00 0.06 10 30 0.00 0.01 0.03 0.55 0.29 0.01 0.15 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.01 0.00 0.01 6 18 0.00 0.01 0.01 0.04 0.02 0.00 0.02 8 24 0.00 0.01 0.02 0.24 0.09 0.00 0.08 10 30 0.00 0.01 0.03 0.73 0.27 0.01 0.19 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.00 0.01 0.00 0.01 6 18 0.00 0.00 0.01 0.03 0.03 0.00 0.02 8 24 0.00 0.01 0.01 0.18 0.10 0.00 0.08 10 30 0.00 0.01 0.03 0.54 0.32 0.01 0.19 TableA.30: Simple-TaskNegotiations: ComparisonofResponseTimesinIndoor-LikeTerrain 304 Comparisonof Comparisonof Comparisonof Robots Targets Capacities SAB SNW SAB(k =1)andSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 10 20 2 290.49 250.54 13.25 100/100 241.50 3.29 92/100 240.32 0.46 94/100 243.27 279.37 276.35 10 30 3 366.68 314.44 13.85 100/100 300.33 4.37 97/100 292.74 2.43 96/100 298.14 353.56 351.04 10 40 4 413.42 364.45 11.58 100/100 347.87 4.39 98/100 338.55 2.59 92/100 342.85 412.74 399.41 10 50 5 457.82 410.74 10.09 100/100 393.04 4.10 94/100 382.78 2.51 92/100 382.43 455.71 449.58 MiniMaxTeamObjective 10 20 2 72.48 41.73 40.04 100/100 40.02 2.55 73/100 38.08 4.19 87/100 43.63 68.65 63.25 10 30 3 89.65 52.22 39.87 100/100 48.04 6.81 83/100 44.89 5.88 87/100 50.72 84.49 78.66 10 40 4 95.29 60.89 34.70 100/100 53.69 10.20 85/100 50.27 5.82 87/100 55.83 92.56 87.18 10 50 5 104.98 64.85 36.69 100/100 59.17 7.53 79/100 55.38 5.59 81/100 59.15 99.88 96.10 MiniLatTeamObjective 10 20 2 413.33 364.29 11.39 100/100 358.93 1.39 93/100 358.37 0.12 95/100 362.29 408.33 403.45 10 30 3 677.06 590.16 12.45 100/100 576.96 2.12 96/100 572.34 0.79 92/100 577.38 669.72 662.66 10 40 4 973.31 843.41 12.98 100/100 825.35 2.08 99/100 817.93 0.87 92/100 830.37 966.27 955.91 10 50 5 1301.26 1135.60 12.32 100/100 1110.14 2.19 91/100 1096.66 1.02 88/100 1110.42 1282.14 1268.62 TableA.31: Simple-TaskNegotiations: ComparisonofTeamCostsforDifferentRobotCapacities Robots Targets Capacities SAB(k = 1) SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 10 20 2 0.00 0.00 0.00 0.03 0.03 0.00 0.01 10 30 3 0.00 0.00 0.03 0.56 0.27 0.01 0.16 10 40 4 0.01 0.01 0.22 5.44 1.77 0.02 0.86 10 50 5 0.01 0.03 1.26 36.78 9.96 0.08 3.41 MiniMaxTeamObjective 10 20 2 0.00 0.00 0.01 0.03 0.03 0.01 0.01 10 30 3 0.00 0.00 0.03 0.67 0.28 0.01 0.18 10 40 4 0.01 0.01 0.28 6.57 1.67 0.02 1.21 10 50 5 0.01 0.03 1.68 47.95 8.46 0.08 3.98 MiniLatTeamObjective 10 20 2 0.00 0.00 0.00 0.02 0.03 0.00 0.01 10 30 3 0.00 0.00 0.03 0.56 0.29 0.01 0.18 10 40 4 0.00 0.01 0.24 6.07 1.88 0.02 1.24 10 50 5 0.01 0.04 1.30 40.58 9.40 0.08 4.05 TableA.32: Simple-TaskNegotiations: ComparisonofResponseTimesforDifferentRobotCapacities Comparisonof Comparisonof Comparisonof Robots Targets Capacities SAB SNW SAB(k =1)andSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 10 20 2 290.49 250.54 13.25 100/100 241.50 3.29 92/100 240.32 0.46 94/100 243.27 279.37 276.35 10 30 3 366.68 314.44 13.85 100/100 300.33 4.37 97/100 292.74 2.43 96/100 298.14 353.56 351.04 10 40 4 413.42 364.40 11.59 100/100 347.42 4.50 98/100 338.55 2.47 91/100 342.85 412.74 399.41 10 50 5 457.82 410.57 10.12 100/100 392.90 4.17 94/100 382.65 2.57 92/100 382.43 455.71 449.58 MiniMaxTeamObjective 10 20 2 72.48 41.73 40.04 100/100 39.97 2.55 73/100 38.04 4.38 87/100 43.63 68.65 63.25 10 30 3 89.65 52.22 39.87 100/100 48.04 6.81 83/100 44.89 5.88 87/100 50.72 84.49 78.66 10 40 4 95.29 60.81 34.80 100/100 53.63 10.39 84/100 50.26 5.89 86/100 55.83 92.56 87.18 10 50 5 104.98 64.82 36.71 100/100 59.14 7.62 79/100 55.23 5.96 79/100 59.15 99.88 96.10 MiniLatTeamObjective 10 20 2 413.33 364.29 11.39 100/100 358.93 1.39 93/100 358.37 0.12 95/100 362.29 408.33 403.45 10 30 3 677.06 590.16 12.45 100/100 576.96 2.12 96/100 572.34 0.79 92/100 577.38 669.72 662.66 10 40 4 973.31 843.09 13.02 100/100 825.21 2.06 98/100 817.91 0.85 92/100 830.37 966.27 955.91 10 50 5 1301.26 1134.06 12.43 100/100 1108.46 2.22 91/100 1096.24 1.23 84/100 1110.42 1282.14 1278.62 TableA.33: Simple-TaskNegotiations: ComparisonofTeamCostsforIdealRobot-CostCalculations 305 Robots Targets Capacities SAB(k = 1) SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 10 20 2 0.00 0.00 0.10 0.13 0.04 0.00 0.01 10 30 3 0.00 0.00 0.11 0.50 0.38 0.01 0.22 10 40 4 0.01 0.01 0.30 4.67 2.52 0.03 1.51 10 50 5 0.02 0.09 3.97 91.33 12.17 0.12 5.86 MiniMaxTeamObjective 10 20 2 0.00 0.00 0.11 0.14 0.04 0.00 0.01 10 30 3 0.00 0.00 0.13 0.67 0.37 0.01 0.22 10 40 4 0.01 0.01 0.36 6.16 2.34 0.03 1.69 10 50 5 0.02 0.10 4.78 92.61 11.37 0.11 6.67 MiniLatTeamObjective 10 20 2 0.00 0.00 0.10 0.13 0.04 0.00 0.01 10 30 3 0.00 0.01 0.12 0.51 0.39 0.01 0.23 10 40 4 0.01 0.01 0.34 5.19 2.57 0.03 1.86 10 50 5 0.02 0.13 5.32 102.29 11.77 0.12 6.71 TableA.34: Simple-TaskNegotiations: ComparisonofResponseTimesforIdealRobot-CostCalculations Comparisonof Comparisonof Comparisonof Robots Targets Random SNW RandomandSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) 2 6 165.62 122.38 25.52 100/100 118.54 2.84 100/100 118.54 0.00 100/100 118.56 125.68 122.14 4 12 328.88 198.87 38.86 100/100 180.04 8.52 98/100 174.34 3.02 100/100 176.42 199.10 191.41 6 18 488.96 259.62 46.46 100/100 228.18 11.52 95/100 220.72 3.03 91/100 222.59 254.96 249.39 8 24 649.51 319.49 50.49 100/100 279.30 12.24 97/100 262.69 5.57 93/100 266.73 312.96 307.91 10 30 809.97 381.36 52.72 100/100 318.62 15.86 96/100 295.42 7.02 93/100 298.14 353.56 351.04 MiniMaxTeamObjective 2 6 94.99 68.83 26.67 100/100 67.39 1.95 100/100 67.39 0.00 100/100 68.44 76.30 71.85 4 12 104.22 65.06 36.65 100/100 58.07 9.72 94/100 56.03 2.96 89/100 57.50 82.64 73.02 6 18 109.32 65.66 39.06 100/100 55.26 14.54 87/100 52.34 4.16 70/100 53.91 83.06 75.16 8 24 112.51 67.43 39.35 100/100 57.62 13.19 86/100 49.94 12.19 83/100 52.36 84.16 77.21 10 30 113.76 70.86 37.23 100/100 58.35 16.37 89/100 51.68 10.06 81/100 50.72 84.49 78.66 MiniLatTeamObjective 2 6 311.60 228.56 26.05 100/100 225.74 1.00 100/100 225.74 0.00 100/100 226.12 231.62 227.80 4 12 617.63 349.71 42.58 100/100 339.47 2.76 98/100 336.75 0.67 98/100 338.53 371.91 363.37 6 18 911.40 457.30 49.35 100/100 438.10 3.81 90/100 430.44 1.66 99/100 434.10 487.30 478.92 8 24 1213.67 558.81 53.66 100/100 527.91 5.14 91/100 519.49 1.49 84/100 527.49 602.95 592.99 10 30 1518.02 620.94 58.91 100/100 587.30 4.99 82/100 574.64 2.08 83/100 577.38 669.72 662.66 TableA.35: Simple-TaskNegotiations: ComparisonofTeamCostsforRandomInitialAllocations 306 Robots Targets Random SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.00 0.00 0.00 6 18 0.00 0.00 0.01 0.07 0.02 0.01 0.02 8 24 0.01 0.01 0.02 0.34 0.09 0.01 0.07 10 30 0.01 0.01 0.05 1.15 0.27 0.01 0.16 MiniMaxTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.00 0.00 0.00 6 18 0.01 0.01 0.01 0.08 0.02 0.00 0.02 8 24 0.01 0.01 0.02 0.41 0.10 0.01 0.07 10 30 0.01 0.01 0.05 1.28 0.28 0.01 0.18 MiniLatTeamObjective 2 6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4 12 0.00 0.00 0.00 0.01 0.00 0.00 0.00 6 18 0.00 0.00 0.01 0.07 0.03 0.01 0.02 8 24 0.01 0.01 0.03 0.33 0.11 0.01 0.08 10 30 0.01 0.01 0.05 1.15 0.29 0.01 0.18 TableA.36: Simple-TaskNegotiations: ComparisonofResponseTimesforRandomInitialAllocations Initial Comparisonof Comparisonof Comparisonof Robots Targets SAB SNW SSIandSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 2 6 159.04 151.04 3.80 88/100 148.85 1.28 97/100 148.67 0.11 99/100 149.51 158.42 152.93 4 12 254.99 229.70 8.96 84/100 222.34 2.85 83/100 220.85 0.63 90/100 221.38 245.24 241.50 6 18 329.44 290.83 11.04 90/100 280.35 3.21 82/100 277.62 0.79 80/100 276.21 319.44 307.24 8 24 400.77 358.56 9.91 89/100 337.67 5.38 81/100 329.21 2.30 77/100 327.99 394.50 384.86 10 30 452.77 398.42 11.53 93/100 373.76 5.75 84/100 363.81 2.40 73/100 370.09 440.39 436.23 MiniMaxTeamObjective 2 6 101.72 89.44 8.96 88/100 86.76 2.46 95/100 86.61 0.15 99/100 87.97 102.71 91.61 4 12 110.34 79.33 24.92 95/100 77.93 0.52 71/100 76.53 0.82 74/100 78.25 104.40 98.64 6 18 113.96 80.02 26.99 91/100 74.89 5.16 68/100 72.58 1.86 69/100 76.39 108.07 98.64 8 24 112.14 81.97 25.22 97/100 74.21 7.14 73/100 70.98 2.76 63/100 74.60 110.66 104.55 10 30 110.54 77.17 26.58 93/100 72.13 4.11 70/100 71.04 -0.20 57/100 73.23 109.69 102.24 MiniLatTeamObjective 2 6 293.06 287.00 1.84 87/100 282.58 1.08 97/100 282.32 0.11 99/100 282.66 288.24 285.88 4 12 465.21 428.05 7.46 84/100 421.59 1.23 88/100 422.07 -0.13 90/100 424.64 461.67 450.63 6 18 606.28 546.72 9.51 90/100 541.97 0.71 67/100 534.49 1.16 87/100 535.15 604.74 594.94 8 24 753.17 656.95 12.14 93/100 648.95 0.98 69/100 640.33 1.20 73/100 648.45 737.09 734.45 10 30 819.38 726.92 10.87 93/100 713.02 1.75 67/100 703.72 1.09 73/100 709.61 821.38 805.58 TableA.37: Simple-TaskNegotiations: ComparisonofTeamCostsinUnknownTerrain 307 Initial Comparisonof Comparisonof Comparisonof Robots Targets SAB SNW SSIandSNW(K=1) SNW SNW(K=1)andSNW(K=2) SNW SNW(K=2)andSNW(K=3) Rollout SAB SAB (k=1) (K=1) Difference(%) Dominance (K=2) Difference(%) Dominance (K=3) Difference(%) Dominance (k=2) (k=3) MiniSumTeamObjective 2 6 16.31 15.65 0.47 84/100 15.24 1.93 96/100 15.19 0.26 99/100 15.29 16.30 15.73 4 12 24.40 22.06 5.80 69/100 21.50 1.21 78/100 21.53 -0.30 81/100 21.27 22.97 22.84 6 18 28.11 25.67 7.10 71/100 24.55 2.86 73/100 24.90 -2.48 73/100 24.23 27.38 26.82 8 24 31.95 28.89 8.12 74/100 27.41 3.19 62/100 26.96 0.19 63/100 26.49 31.73 31.19 10 30 33.42 29.85 9.76 77/100 27.90 5.73 67/100 27.47 0.80 68/100 27.74 32.76 31.69 MiniMaxTeamObjective 2 6 16.12 15.92 -7.09 72/100 15.44 2.90 92/100 15.44 -0.04 98/100 15.87 16.95 15.42 4 12 24.28 21.92 6.87 74/100 21.88 -3.23 58/100 22.08 -2.90 66/100 22.16 24.19 23.54 6 18 28.84 26.53 5.08 65/100 26.04 0.35 58/100 25.20 1.56 64/100 27.09 28.85 27.48 8 24 32.96 31.26 2.78 60/100 29.62 3.13 64/100 29.94 -3.25 50/100 30.09 32.54 32.21 10 30 34.00 31.99 4.56 65/100 30.53 2.58 59/100 30.68 -2.07 55/100 32.37 34.46 34.90 MiniLatTeamObjective 2 6 16.09 15.78 -0.90 83/100 15.43 1.56 98/100 15.38 0.31 99/100 15.40 16.11 15.80 4 12 24.57 22.08 7.35 77/100 21.29 1.95 83/100 21.73 -2.97 86/100 21.94 24.48 23.40 6 18 28.46 25.91 6.88 63/100 25.82 -1.45 65/100 25.17 1.12 80/100 24.80 28.82 28.42 8 24 33.47 28.91 11.43 74/100 27.69 2.96 73/100 27.47 0.14 68/100 28.27 32.45 32.21 10 30 33.83 28.94 13.09 81/100 28.54 0.37 64/100 27.91 1.28 67/100 28.49 35.01 34.08 TableA.38: Simple-TaskNegotiations: ComparisonofNumbersofRepeatedExecutionsinUnknownTerrain Robots Targets SAB(k = 1) SNW(K=1) SNW(K=2) SNW(K=3) Rollout SAB(k=2) SAB(k=3) MiniSumTeamObjective 2 6 0.04 0.04 0.12 0.15 0.05 0.02 0.02 4 12 0.12 0.15 0.28 0.22 0.14 0.06 0.07 6 18 0.18 0.22 0.38 0.54 0.33 0.09 0.18 8 24 0.23 0.45 0.58 1.95 0.95 0.15 0.48 10 30 0.29 0.57 0.74 5.33 2.21 0.19 1.07 MiniMaxTeamObjective 2 6 0.03 0.04 0.12 0.15 0.05 0.02 0.02 4 12 0.08 0.16 0.32 0.23 0.15 0.07 0.09 6 18 0.16 0.22 0.42 0.74 0.37 0.13 0.24 8 24 0.23 0.47 0.65 3.10 1.08 0.21 0.59 10 30 0.29 0.63 0.82 10.02 2.71 0.26 1.10 MiniLatTeamObjective 2 6 0.03 0.05 0.12 0.15 0.05 0.02 0.02 4 12 0.09 0.17 0.25 0.22 0.14 0.07 0.09 6 18 0.16 0.23 0.37 0.5 0.32 0.13 0.22 8 24 0.24 0.49 0.53 1.67 0.90 0.19 0.48 10 30 0.28 0.56 0.72 4.77 2.03 0.26 0.93 TableA.39: Simple-TaskNegotiations: ComparisonofCumulativeResponseTimesinUnknownTerrain 308 TableA.22tabulatestheteamcostsofourtask-allocationalgorithmswiththeopti- malalgorithminoutdoor-liketerrainwithasmallsamplesize(15instances). Theinitial allocationsfortheSNW(K =1,2,3)algorithmsareproducedbytheSAB(k =1)algo- rithm. Wemakethefollowingobservations: • TheteamcostsoftheSNW(K =1)algorithmaresmallerthanthoseoftheinitial allocations (no matter what the team objective is). For the MiniSum team objec- tive, the SNW (K = 1) algorithm reduces the team costs of the initial allocations by 10.23 percent on average. For the MiniMax team objective, the SNW (K = 1) algorithm reduces the team costs of the initial allocations by 32.97 percent on average. FortheMiniLatteamobjective,theSNW(K =1)algorithmreducesthe teamcostsoftheinitialallocationsby10.68percentonaverage. Thisobservation meetsourexpectationsincetheSNW(K =1)algorithmconstructsandperforms profitable task exchanges between (two) robots (if any) to improve the current allocations. It is guaranteed that the SNW (K = 1) algorithm does not increase the team costs of the initial allocations. However, the team costs of the SNW (K = 1) algorithm are still larger than those of the rollout algorithm (no matter what theteamobjectiveis). ThisobservationmeetsourexpectationsincetheSNW(K =1)constructsandperformsonlysingletaskexchangesamongrobotstoimprove thecurrentallocations. • TheteamcostsoftheSNW(K =2)algorithmaresmallerthanthoseoftheSNW (K =1)algorithm, andtheteamcostsoftheSNW(K =3)algorithmaresmaller than those of the SNW (K = 2) algorithm (no matter what the team objective is). For the MiniSum team objective, the SNW (K = 2) algorithm reduces the team costsoftheSNW(K =1)algorithmonaverageby2.87percent,andtheSAB(K = 3) algorithm reduces the team costs of the SNW (K = 1) algorithm on average by 1.37 percent. For the MiniMax team objective, the SNW (K = 2) algorithm 309 reducedtheteamcostsoftheSNW(K =1)algorithmonaverageby7.22percent, andtheSNW(K =3)algorithmreducestheteamcostsoftheSNW(K =2)algo- rithmonaverageby3.34percent. FortheMiniLatteamobjective,theSNW(K = 2)algorithmreducestheteamcostsoftheSNW(K =1)algorithmonaverageby 1.47percent,andtheSNW(K =3)algorithmreducestheteamcostsoftheSNW (K = 2) algorithm on average by 0.67 percent. One-sided binomial tests [CB90] withnormalapproximation(n = 100;p = 0.5;σ 2 = 50)areperformedtosupport the hypothesis that the team costs of SNW algorithms with a larger number ofK are not larger than those of SNW algorithms with a smaller number of K with confidence 0.9995 (which is significant). This observation meets our expectation sinceSNWalgorithmswithalargernumberofK constructandperformprofitable task exchanges that involve more tasks and robots than SNW algorithms with a smallernumberofK. • TheteamcostsofSNW(K=2,3)algorithmsareveryclosetothoseoftherollout algorithm and smaller than those of the SAB (k = 2, 3) algorithms (no matter what the team objective is). For example, the team costs of the SNW (K = 2, 3) algorithms are consistently smaller than those of the rollout algorithm for the MiniMax team objective. These results justify our generalization of single task exchanges to sequential negotiations with K-swaps and demonstrate that SNW algorithms with a small number of K can reduce the team costs of the initial allocationseffectively. TableA.23tabulatestheteamcostsofourtask-allocationalgorithms(exceptforthe optimal algorithm) in outdoor-like terrain with a larger sample size (100 instances). TheseresultssupportourpreviousstatementthatSNWalgorithmswithasmallnumber ofK canreducetheteamcostoftheinitialallocationseffectively. 310 r 1 x 1 x 3 r 2 x 4 x 2 x 5 x 6 r 3 FigureA.3: Example8ofMulti-RobotRouting r 1 x 1 x 3 r 2 x 4 x 2 x 5 x 6 r 3 OptimalAllocations (TeamCost=8) r 1 x 1 x 3 r 2 x 4 x 2 x 5 x 6 r 3 InitialAllocations (TeamCost=13) FigureA.4: AllocationsforExampleinFigureA.3(MiniMaxTeamObjective) TableA.24andA.25tabulatetheamountofrobotcommunicationoftheSNW(K =1,2,3)algorithmsinoutdoor-liketerrainwith100instances. Wecomparethenumber of profitable partial multi-swaps robots have to send with the number of total partial- swaps robots can construct before each iteration of the proposal-construction phase for all rounds of the SNW (K = 1, 2, 3) algorithms. We also compare the number of profitable complete multi-swaps robots have to send with the number of total complete multi-swapsrobotscanconstructduringallroundsoftheSNW(K =1,2,3)algorithms. Wemakethefollowingobservations: • The number of profitable partial (and complete) multi-swaps increases when the problem size increases (no matter what the team objective is). This observation 311 Robots PartialZero-SwapsBeforeIteration1 Loss(Profitable) (OutSwaps,InSwaps) MiniSum MiniMax MiniLat r 1 ({(r 1 ,−,x 1 ,−)}, {(r 1 ,−,−,x 2 )}) 3 (no) -7 (no) 3 (no) r 1 ({(r 1 ,−,x 1 ,−)}, {(r 1 ,−,−,x 3 )}) 1 (no) -9 (no) 1 (no) r 1 ({(r 1 ,−,x 1 ,−)}, {(r 1 ,−,−,x 4 )}) 3 (no) -7 (no) 3 (no) r 1 ({(r 1 ,−,x 1 ,−)}, {(r 1 ,−,−,x 6 )}) 1 (no) -9 (no) 1 (no) r 1 ({(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 2 )}) 3 (no) -7 (no) 5 (no) r 1 ({(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 3 )}) 3 (no) -7 (no) 5 (no) r 1 ({(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 4 )}) 8 (no) -2 (no) 10 (no) r 1 ({(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 6 )}) 4 (no) -6 (no) 5 (no) r 1 ({(r 1 ,−,x 1 ,−),(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 2 ),(r 1 ,−,−,x 3 )}) 7 (no) 1 (no) 10 (no) r 1 ({(r 1 ,−,x 1 ,−),(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 2 ),(r 1 ,−,−,x 4 )}) 9 (no) -1 (no) 12 (no) r 1 ({(r 1 ,−,x 1 ,−),(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 2 ),(r 1 ,−,−,x 6 )}) 5 (no) -5 (no) 6 (no) r 1 ({(r 1 ,−,x 1 ,−),(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 3 ),(r 1 ,−,−,x 4 )}) 9 (no) -1 (no) 12 (no) r 1 ({(r 1 ,−,x 1 ,−),(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 3 ),(r 1 ,−,−,x 6 )}) 5 (no) -5 (no) 6 (no) r 1 ({(r 1 ,−,x 1 ,−),(r 1 ,−,x 5 ,−)}, {(r 1 ,−,−,x 4 ),(r 1 ,−,−,x 6 )}) 1 (no) -9 (no) 2 (no) r 2 ({(r 2 ,−,x 3 ,−)}, {(r 2 ,−,−,x 1 )}) 0 (no) 0 (no) 1 (no) r 2 ({(r 2 ,−,x 3 ,−)}, {(r 2 ,−,−,x 2 )}) 0 (no) 0 (no) 0 (no) r 2 ({(r 2 ,−,x 3 ,−)}, {(r 2 ,−,−,x 5 )}) 0 (no) 0 (no) 3 (no) r 2 ({(r 2 ,−,x 3 ,−)}, {(r 2 ,−,−,x 6 )}) 0 (no) 0 (no) 6 (no) r 2 ({(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 1 )}) -5 (yes) -5 (yes) -5 (yes) r 2 ({(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 2 )}) -2 (yes) -2 (yes) -2 (yes) r 2 ({(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 5 )}) -5 (yes) -5 (yes) -5 (yes) r 2 ({(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 6 )}) -2 (yes) -2 (yes) -2 (yes) r 2 ({(r 2 ,−,x 3 ,−),(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 1 ),(r 2 ,−,−,x 2 )}) -5 (yes) -5 (yes) -5 (yes) r 2 ({(r 2 ,−,x 3 ,−),(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 1 ),(r 2 ,−,−,x 5 )}) -5 (yes) -5 (yes) -4 (yes) r 2 ({(r 2 ,−,x 3 ,−),(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 1 ),(r 2 ,−,−,x 6 )}) -2 (yes) -2 (yes) -1 (yes) r 2 ({(r 2 ,−,x 3 ,−),(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 2 ),(r 2 ,−,−,x 5 )}) -3 (yes) -3 (yes) -3 (yes) r 2 ({(r 2 ,−,x 3 ,−),(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 2 ),(r 2 ,−,−,x 6 )}) -2 (yes) -2 (yes) -2 (yes) r 2 ({(r 2 ,−,x 3 ,−),(r 2 ,−,x 4 ,−)}, {(r 2 ,−,−,x 5 ),(r 2 ,−,−,x 6 )}) -2 (yes) -2 (yes) 1 (no) r 3 ({(r 3 ,−,x 2 ,−)}, {(r 3 ,−,−,x 1 )}) -2 (yes) -7 (no) -3 (yes) r 3 ({(r 3 ,−,x 2 ,−)}, {(r 3 ,−,−,x 3 )}) 2 (no) -3 (no) 4 (no) r 3 ({(r 3 ,−,x 2 ,−)}, {(r 3 ,−,−,x 4 )}) 0 (no) -5 (no) 4 (no) r 3 ({(r 3 ,−,x 2 ,−)}, {(r 3 ,−,−,x 5 )}) -2 (yes) -7 (no) -1 (yes) r 3 ({(r 3 ,−,x 6 ,−)}, {(r 3 ,−,−,x 1 )}) -4 (yes) -9 (no) -5 (yes) r 3 ({(r 3 ,−,x 6 ,−)}, {(r 3 ,−,−,x 3 )}) 0 (no) -5 (no) 0 (no) r 3 ({(r 3 ,−,x 6 ,−)}, {(r 3 ,−,−,x 4 )}) 2 (no) -3 (no) 2 (no) r 3 ({(r 3 ,−,x 6 ,−)}, {(r 3 ,−,−,x 5 )}) -1 (yes) -6 (no) -1 (yes) r 3 ({(r 3 ,−,x 2 ,−),(r 2 ,−,x 6 ,−)}, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 3 )}) -4 (yes) -9 (no) -5 (yes) r 3 ({(r 3 ,−,x 2 ,−),(r 2 ,−,x 6 ,−)}, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 4 )}) 0 (no) -5 (no) -1 (yes) r 3 ({(r 3 ,−,x 2 ,−),(r 2 ,−,x 6 ,−)}, {(r 3 ,−,−,x 1 ),(r 3 ,−,−,x 5 )}) -5 (yes) -10 (no) -6 (yes) r 3 ({(r 3 ,−,x 2 ,−),(r 2 ,−,x 6 ,−)}, {(r 3 ,−,−,x 3 ),(r 3 ,−,−,x 4 )}) 4 (no) -1 (no) 6 (no) r 3 ({(r 3 ,−,x 2 ,−),(r 2 ,−,x 6 ,−)}, {(r 3 ,−,−,x 3 ),(r 3 ,−,−,x 5 )}) -1 (yes) -6 (no) 1 (no) r 3 ({(r 3 ,−,x 2 ,−),(r 2 ,−,x 6 ,−)}, {(r 3 ,−,−,x 4 ),(r 3 ,−,−,x 5 )}) 0 (no) -5 (no) 1 (no) TableA.40: PartialZero-SwapsEvaluatedBeforeIteration1forExampleinFigureA.3 meets our expectation since the number of possible task exchanges, and thus the numberoftheprofitableones,increaseswhentherearemorerobotsandtargets. • The number of profitable partial multi-swaps increases when the number ofK in SNW algorithms increases (no matter what the team objective is). This obser- vation meets our expectation since SNW algorithms with a larger number of K 312 Robots PartialOne-SwapsBeforeIteration2 Loss(Profitable) (OutSwaps,ExchangeSwaps,InSwaps) MiniSum MiniMax MiniLat r 1 (∅, {(r 1 ,r 2 ,x 1 ,x 4 )}, ∅) -2 (yes) -5 (yes) -2 (yes) r 1 ({(r 1 ,−,x 5 ,−)}, {(r 1 ,r 2 ,x 1 ,x 4 )}, {(r 1 ,−,−,x 2 )}) 4 (no) -1 (yes) 7 (no) r 1 ({(r 1 ,−,x 5 ,−)}, {(r 1 ,r 2 ,x 1 ,x 4 )}, {(r 1 ,−,−,x 6 )}) -4 (yes) -5 (yes) -3 (yes) r 1 ({(r 2 ,−,x 3 ,−)}, {(r 1 ,r 2 ,x 1 ,x 4 )}, {(r 2 ,−,−,x 2 )}) -2 (yes) -5 (yes) -2 (yes) r 1 ({(r 2 ,−,x 4 ,−)}, {(r 1 ,r 2 ,x 1 ,x 3 )}, {(r 2 ,−,−,x 2 )}) -4 (yes) -5 (yes) -4 (yes) r 1 ({(r 2 ,−,x 3 ,−)}, {(r 1 ,r 2 ,x 1 ,x 4 )}, {(r 2 ,−,−,x 6 )}) 1 (no) -3 (yes) 2 (no) r 1 ({(r 2 ,−,x 4 ,−)}, {(r 1 ,r 2 ,x 1 ,x 3 )}, {(r 2 ,−,−,x 6 )}) -1 (yes) -2 (yes) 0 (no) r 1 (∅, {(r 1 ,r 2 ,x 5 ,x 4 )}, ∅) 3 (no) -2 (yes) 5 (no) r 1 ({(r 1 ,−,x 1 ,−)}, {(r 1 ,r 2 ,x 5 ,x 4 )}, {(r 1 ,−,−,x 2 )}) 4 (no) -1 (yes) 7 (no) r 1 ({(r 1 ,−,x 1 ,−)}, {(r 1 ,r 2 ,x 5 ,x 4 )}, {(r 1 ,−,−,x 6 )}) -4 (yes) -5 (yes) -3 (yes) r 1 ({(r 2 ,−,x 3 ,−)}, {(r 1 ,r 2 ,x 5 ,x 4 )}, {(r 2 ,−,−,x 2 )}) 5 (no) -2 (yes) 7 (no) r 1 ({(r 2 ,−,x 4 ,−)}, {(r 1 ,r 2 ,x 5 ,x 3 )}, {(r 2 ,−,−,x 2 )}) 0 (no) -3 (yes) 2 (no) r 1 ({(r 2 ,−,x 3 ,−)}, {(r 1 ,r 2 ,x 5 ,x 4 )}, {(r 2 ,−,−,x 6 )}) 6 (no) -2 (yes) n/a r 1 ({(r 2 ,−,x 4 ,−)}, {(r 1 ,r 2 ,x 5 ,x 3 )}, {(r 2 ,−,−,x 6 )}) 1 (no) -2 (yes) n/a r 3 (∅, {(r 3 ,r 2 ,x 2 ,x 4 )}, ∅) -2 (yes) -2 (yes) 2 (no) r 3 ({(r 3 ,−,x 6 ,−)}, {(r 3 ,r 2 ,x 2 ,x 4 )}, {(r 3 ,−,−,x 1 )}) -2 (yes) -2 (yes) -3 (yes) r 3 ({(r 3 ,−,x 6 ,−)}, {(r 3 ,r 2 ,x 2 ,x 4 )}, {(r 3 ,−,−,x 5 )}) -2 (yes) -2 (yes) -1 (yes) r 3 ({(r 2 ,−,x 3 ,−)}, {(r 3 ,r 2 ,x 2 ,x 4 )}, {(r 2 ,−,−,x 1 )}) -5 (yes) -5 (yes) -1 (yes) r 3 ({(r 2 ,−,x 4 ,−)}, {(r 3 ,r 2 ,x 2 ,x 3 )}, {(r 2 ,−,−,x 1 )}) -3 (yes) -3 (yes) -1 (yes) r 3 ({(r 2 ,−,x 3 ,−)}, {(r 3 ,r 2 ,x 2 ,x 4 )}, {(r 2 ,−,−,x 5 )}) -3 (yes) -3 (yes) 1 (no) r 3 ({(r 2 ,−,x 4 ,−)}, {(r 3 ,r 2 ,x 2 ,x 3 )}, {(r 2 ,−,−,x 5 )}) 0 (no) -3 (yes) 1 (no) r 3 (∅, {(r 3 ,r 2 ,x 6 ,x 4 )}, ∅) 0 (no) -2 (yes) 0 (no) r 3 ({(r 3 ,−,x 2 ,−)}, {(r 3 ,r 2 ,x 6 ,x 4 )}, {(r 3 ,−,−,x 1 )}) -2 (yes) -2 (yes) -3 (yes) r 3 ({(r 3 ,−,x 2 ,−)}, {(r 3 ,r 2 ,x 6 ,x 4 )}, {(r 3 ,−,−,x 5 )}) -2 (yes) -2 (yes) 1 (no) r 3 ({(r 2 ,−,x 3 ,−)}, {(r 3 ,r 2 ,x 6 ,x 4 )}, {(r 2 ,−,−,x 1 )}) 0 (no) -2 (yes) 1 (no) r 3 ({(r 2 ,−,x 4 ,−)}, {(r 3 ,r 2 ,x 6 ,x 3 )}, {(r 2 ,−,−,x 1 )}) -3 (yes) -3 (yes) -3 (yes) r 3 ({(r 2 ,−,x 3 ,−)}, {(r 3 ,r 2 ,x 6 ,x 4 )}, {(r 2 ,−,−,x 5 )}) 0 (no) -2 (yes) n/a r 3 ({(r 2 ,−,x 4 ,−)}, {(r 3 ,r 2 ,x 6 ,x 3 )}, {(r 2 ,−,−,x 5 )}) -2 (yes) -2 (yes) n/a TableA.41: PartialOne-SwapsEvaluatedBeforeIteration2forExampleinFigureA.3 construct and perform task exchanges involving more robots and targets in each round. • The number of profitable partial multi-swaps before the first iteration of the proposal-construction phase for the MiniMax team objective is smaller than or close to those for the MiniSum and MiniLat team objectives. This observation meets our expectation since the definition of a profitable partial multi-swap for the MiniMax team objective is different from that for the MiniSum and MiniLat team objectives: A profitable partial multi-swap for the MiniMax team objective has to contain at least one robot whose current robot cost is equal to the current teamcost. FigureA.4showsamulti-robotroutingexamplewithsimpletargetsfor the MiniMax team objective. There exists only one capability. Robotsr 1 ,r 2 and r 3 providethiscapability,andtargetsx 1 ,x 2 ,x 3 ,x 4 ,x 5 andx 6 requireit. Therobot 313 capacityofallrobotsis2sothattherobotcapacity-constraintistightinthisexam- ple. The optimal allocations (X r 1 ={x 4 ,x 6 },X r 2 ={x 3 ,x 5 },X r 3 ={x 1 ,x 2 }) with the smallest team cost 8 are shown in the left part of Figure A.4. The initial allocations of this example (X r 1 ={x 1 ,x 5 },X r 2 ={x 3 ,x 4 },X r 3 ={x 2 ,x 6 }) with team cost 13 are shown in the right part of Figure A.4. Table A.40 tabulates thepartialzero-swapsrobotsconstructandevaluatebeforethefirstiterationofthe proposal-construction phase of the first round of the SNW (K = 2) algorithm for all three team objectives. 1 The number of profitable zero-swaps for the MiniSum and MiniLat team objectives is larger than that for the MiniMax team objective sinceonlythepartialzero-swapsofrobotr 2 haveachancetobeprofitableinthis example. • The number of profitable partial multi-swaps after the first iteration of the proposal-construction phase for the MiniMax team objective is larger than those for the MiniSum and MiniLat team objectives. This observation is surprising but canbe explainedasfollows: The lossof a partialmulti-swaps k isdetermined by thelargestrobotcostofrobotsinR(s k )fortheMiniMaxteamobjective. Thus, a partialmulti-swapisprofitableaslongasthelargestrobotcostissmallerthanthe current team cost after performing this partial multi-swap for the MiniMax team objective,eventhoughtherobotcostsofsomerobotsafterperformingthispartial multi-swap may increase a lot (as long as they are still smaller than the current teamcost). Consideragainthemulti-robotroutingexampleshowninFigureA.3. Table A.41 tabulates the partial one-swaps robotsr 1 andr 2 need to construct and evaluate before the second iteration of the proposal-construction stage of the first 1 Sincetherobotcapacity-constraintistightinthisexample,wedonotincludeanypartialmulti-swaps in which the number of targets moved to a robot is different from the number of targets moved from that robot. 314 round of the SNW (K = 2) algorithm for all three team objectives. It shows that allpartialone-swapsthatmovetargetx 4 fromrobotr 2 areprofitablefortheMini- Maxteamobjective. Thenumberofprofitablepartialone-swapsfortheMiniSum and MiniLat team objective is about half that of the MiniMax team objective in thisexample. • The number of profitable partial multi-swaps robots need to send is only a small part of the partial multi-swaps robots can construct in the SNW (K = 1, 2, 3) algorithms (no matter what the team objective is). For example, the number of profitable partial multi-swaps is consistently less than 5 percent of the number of partial multi-swaps in the SNW (K = 1, 2, 3) algorithms for all three team objectiveswhenthereare10robotsand30targets. Table A.26 tabulates the response times of our task-allocation algorithms (except for the optimal algorithm) in outdoor-like terrain with 100 instances. We make the followingobservations: • TheresponsetimesofallSNW(K =1,2,3)algorithmsincreasewiththeproblem size (no matter what the team objective is). This observation meets our expecta- tion since there are more partial multi-swaps robots need to construct and send whentherearemorerobotsandtargets. • The response times of all SNW (K = 1, 2) algorithms are very small (no matter what the team objective is). This observation meets our expectation since the number of partial multi-swaps that robots need to construct and send is small for the SNW (K = 1, 2) algorithms. For example, the number of all possible partial multi-swapsisquadraticinthenumberoftargetsfortheSNW(K =1)algorithm. These results demonstrate that SNW algorithms with a small number of K can re-allocatetasksamongrobotsefficiently. 315 • The response times of the SNW (K = 3) algorithm are larger than those of the other task-allocation algorithms and increase quickly with the problem size (no matter what the team objective is). This observation meets our expectation since the number of partial multi-swaps that robots need to construct and send for the SNW (K = 3) algorithm are much larger than those for the SNW (K = 1, 2) algorithms,asshowninTableA.25. • The response times of the SNW (K = 3) algorithm for the MiniMax team objec- tive is larger that those for the MiniSum and MiniLat team objectives. This observation meets our expectation since there are more profitable multi-swaps that robots need to construct and send for the MiniMax team objective, as shown inTableA.25. Tables A.27 and A.29 tabulate the team costs of our task-allocation algorithms in emptyandindoor-liketerrain,respectively. Wemakethefollowingobservations: • The team costs of our task-allocation algorithms in empty terrain are all smaller than those in outdoor-like terrain, and the team costs of our task-allocation algo- rithms in indoor-like terrain are all larger than those in outdoor-like terrain (no matter what the team objective is). This observation meets our expectation since robotsinemptyterraindonothavetocircumnavigateanyblockedcellsandrobots in indoor-like terrain have to go through doors at fixed positions to visit targets, whereas they only have to circumnavigate groups of randomly blocked cells in outdoor-liketerrain. • TheteamcostsoftheSNW(K =1)algorithmaresmallerthanthoseoftheinitial allocations in both empty and indoor-liketerrain (no matter what the team objec- tive is). The team costs of the SNW (K = 2, 3) algorithms are very close to or 316 even smaller than those of the rollout algorithm (no matter what the team objec- tive is). These results demonstrate that the SNW algorithms with a small number ofK reduce the team costs of the initial allocations effectively in different kinds ofterrain. TablesA.28andA.30tabulatetheresponsetimesofourtask-allocationalgorithms inemptyandindoor-liketerrain,respectively. Theresultsarealmostidenticaltothose in outdoor-like terrain, as shown in Table A.26, and demonstrate that the SNW algo- rithmswithasmallnumberofK re-allocatetargetsamongrobotsefficientlyindifferent kindsofterrain. TableA.31tabulatestheteamcostsofourtask-allocationalgorithmsinoutdoor-like terrainfordifferentrobotcapacities. Wemakethefollowingobservations: • The team costs of all task-allocation algorithms increase when the robot capac- ity increases (no matter what the team objective is). This observation meets our expectationsincetherobotcostincreaseswhenarobotneedstovisitmoretargets. • TheteamcostsoftheSNW(K =1)algorithmareconsistentlysmallerthanthose of the initial allocations for different robot capacities (no matter what the team objective is). The team costs of the SNW (K = 2, 3) algorithms are consistently close to or smaller than those of the rollout algorithm for different robot capaci- ties (no matter what the team objective is). These results demonstrate that SNW algorithms with a small number of K can reduce the team costs of the initial allocationseffectivelyfordifferentrobotcapacities. Table A.32 tabulates the response times of our task-allocation algorithms in outdoor-liketerrainfordifferentrobotcapacities. Wemakethefollowingobservations: • The response times of the SNW (K = 1, 2) algorithms remain small when the robot capacity increases. For example, the response times of the SNW (K = 2) 317 algorithmarelessthantwosecondsinourexperiments. Theseresultsdemonstrate that SNW algorithms with a small number of K can re-allocate targets among robotsefficientlyfordifferentrobotcapacities. • The response times of the SNW (K = 3) algorithm increase dramatically when the robot capacity increases. For example, the response times of the SNW (K = 3) algorithm are several times larger than those of the rollout algorithm when the robot capacity is 5. This observation is not surprising since the number of partial multi-swaps robots needs to construct and send for the SNW (K = 3) algorithm increases faster than that for the SNW (K = 1, 2) algorithms when each robot is able to visit more targets. These results demonstrate that SNW algorithms with a large number of K may not respect the real-time constraint when the robot capacityincreases. TableA.33tabulatestheteamcostsofourtask-allocationalgorithmsinoutdoor-like terrainforidealrobot-costcalculations. Wemakethefollowingobservations: • TheteamcostsoftheSNW(K =1,2,3)algorithmswithidealrobot-costcalcula- tionsareslightlysmallerthanthosewithapproximatedrobot-costcalculations,as showninTableA.31,whentherobotcapacityislargerthan3(nomatterwhatthe teamobjectiveis). Thisobservationmeetsourexpectationsincetheapproximated robotcostscalculatedbythetwo-optheuristicarenotsmallerthantheexactones. • TheteamcostsoftheSNW(K =1,2,3)algorithmswithidealrobot-costcalcula- tionsareveryclosetothosewithapproximatedrobot-costcalculations(nomatter what the team objective is). This observation demonstrate that the approximated robot costs calculated by the two-opt heuristic are very close to the exact ones in ourexperiments. 318 Table A.34 tabulates the response times of our task-allocation algorithms in outdoor-like terrain for ideal robot-cost calculations. We make the following obser- vations: • The response times of the SNW (K = 1, 2, 3) algorithms with ideal robot-cost calculationsarelargerthanthosewithapproximatedrobot-costcalculations. This observation meets our expectation since a robot needs to evaluate all possible routesthatvisititsallocatedtargetstodeterminetheexactrobotcost. • The response times of the SNW (K = 1, 2, 3) algorithms with ideal robot-cost calculationsincreasesdramaticallywhentherobotcapacityincreases. Thisobser- vation meets our expectation since the number of routes that a robot with ideal robot-cost calculations needs to evaluate is exponential in the number of its allo- catedtargets. TableA.35tabulatestheteamcostsofourtask-allocationalgorithmsinoutdoor-like terrain if the initial allocations are produced by the randomized algorithm. We make thefollowingobservations: • The team costs of the SNW (K = 1, 2, 3) algorithms are smaller than those of the initial allocations produced by the randomized algorithm (no matter what the team objective is). For the MiniSum team objective, the SNW (K = 1) algorithm reduces the team costs of the initial allocations on average by 42.81 percent, the SNW(K =2)algorithmreducestheteamcostsonaverageby10.20percentcom- pared to the SNW (K = 1) algorithm, and the SNW (K = 3) algorithm reduces the team costs on average by 3.73 percent compared to the SNW (K = 2) algo- rithm. For the MiniMax team objective, the SNW (K = 1) algorithm reduces the team costs of the initial allocations on average by 35.79 percent, the SNW (K = 2) algorithm reduces the team costs on average by 11.15 percent compared to 319 the SNW (K = 1) algorithm, and the SNW (K = 3) algorithm reduces the team costs on average by 5.87 percent compared to the SNW (K = 2) algorithm. For the MiniLat team objective, the SNW (K = 1) algorithm reduces the team costs oftheinitialallocationsonaverageby46.11percent,theSNW(K =2)algorithm reducestheteamcostsonaverageby3.54percentcomparedtotheSNW(K =1) algorithm, and theSNW (K = 3)algorithm reduces theteam costs onaverageby 1.18percentcomparedtotheSNW(K =2)algorithm. • The team costs of the SNW (K = 1, 2, 3) algorithms when the initial allocations are produced by the randomized algorithm are larger than those when the initial allocations are produced by the SAB (k = 1) algorithm, as shown in Table A.23 (no matter what the team objective is). For example, the team costs of the SNW (K = 1) algorithm when the initial allocations are produced by the randomized algorithm are on average 10.96 percent larger than the team costs of the SNW (K = 1) algorithm when the initial allocations are produced by the SAB (k = 1) algorithm. However, the difference decreases when the number of K increases. Forexample,thedifferenceintheteamcostsoftheSNW(K =2)algorithmwhen theinitialallocationsproducedbytwodifferenttask-allocationalgorithmsisonly 4.47 percent on average and continues to drop to 2.02 percent on average for the SNW(K =3)algorithm. • The team costs of the SNW (K = 2, 3) algorithms when the initial allocations are produced by the randomized algorithm are very close to those of the rollout algorithm (no matter what the team objective is). These results demonstrate that SNWalgorithmswithasmallnumberofK canreducetheteamcostsoftheinitial allocationseffectivelynomatterhowtheinitialallocationsareproduced. 320 Table A.36 tabulates the response times of our task-allocation algorithms in outdoor-liketerrainiftheinitialallocationsareproducedbytherandomizedalgorithm. Wemakethefollowingobservations: • The response times of the SNW (K = 1, 2, 3) algorithms remain small when the initial allocations are produced by the randomized algorithm (no matter what the team objective is). For example, the response times of the SNW (K = 2) algorithm are consistently less than 0.1 seconds. This observation demonstrate thatSNWalgorithmswithasmallnumberofK canre-allocatetasksamongrobots efficientlynomatterhowtheinitialallocationsareproduced. • TheresponsetimesoftheSNW(K=2,3)algorithmswhentheinitialallocations are produced by the randomized algorithm are larger than those when the initial allocations are produced by the SAB (k = 1) algorithm, as shown in Table A.26 (no matter what the team objective is). This observation meets our expectation since the number of profitable partial multi-swaps that robots need to construct andsendismuchlargerwhentheinitialallocationsaremoresub-optimal. Table A.37 tabulates the team costs of our task-allocation algorithms in unknown outdoor-liketerrain. Wemakethefollowingobservations: • TheteamcostsoftheSNW(K =1)algorithminunknownterrainaresmallerthan thoseoftheSAB(k =1)algorithmthatproducestheinitialallocations(nomatter what the team objective is). This observation meets our expectation since the SNW (K = 1) algorithm can reduce the team costs of the SAB (k = 1) algorithm in known terrain, as shown in Table A.23. Statistical tests support the hypothesis that the team costs of the SNW (K = 1) algorithm are not larger than those of the SAB (k = 1) algorithm in unknown terrain with confidence 0.9995 (which is significant). 321 • The team costs of SNW algorithms with a larger number of K in unknown ter- rain are smaller than those of SNW algorithms with a smaller number of K (no matter what the team objective is). This observation meets our expectation since SNW algorithms with a larger number of K can reduce the team costs of SNW algorithmswithasmallernumberofK inknownterrain,asshowninTableA.23. StatisticaltestssupportthehypothesisthattheteamcostsofSNWalgorithmswith alargernumberofK arenotlargerthanthoseofSNWalgorithmswithasmaller numberofK inunknownterrainwithconfidence0.9995(whichissignificant). Table A.38 tabulates the numbers of repeated executions of our task-allocation algorithmsinunknownoutdoor-liketerrain. Wemakethefollowingobservations: • The numbers of repeated executions of the SNW (K = 1) algorithm in unknown terrain are smaller than those of the SAB (k = 1) algorithm (no matter what the teamobjectiveis). Thisobservationmeetsourexpectationsincetheteamcostsof theSNW(K =1)algorithmaresmallerthanthoseoftheSAB(k=1)algorithmin unknownterrain. Thus,thechancethatthereareblockedcellsintherobotroutes fortheSNW(K =1)algorithmissmallerthanthatfortheSAB(k=1)algorithm. Statistical tests support the hypothesis that the number of repeated executions of theSNW(K =1)algorithmarenolargerthanthoseoftheSAB(k =1)algorithm inunknownterrainwithconfidence0.9995(whichissignificant). • The numbers of repeated executions of SNW algorithms with a larger number of K in unknown terrain tend to be smaller than those of SNW algorithms with a smaller number ofK (no matter what team objective is). This observation meets our expectation since the team costs of SNW algorithms with a larger number of K in unknown terrain are smaller than those of SNW algorithms with a smaller numberofK. However,statisticaltestssupportthehypothesisthatthenumbersof 322 repeatedexecutionsof SNWalgorithms witha largernumber ofK arenotlarger than those of SNW algorithms with a smaller number ofK only with confidence 0.904(whichisnotsignificant). Table A.39 tabulates the cumulative response times of our task-allocation algo- rithmsinunknownterrain. Wemakethefollowingobservations: • ThecumulativeresponsetimesoftheSNW(K =1)algorithminunknownterrain are larger than those of the SAB (k = 1) algorithm (no matter what the team objective is). This observation meets our expectation since the response times of theSNW(K =1)algorithmarelargerthanthoseoftheSAB(k =1)algorithmin knownterrain. • The cumulative response times of SNW algorithms with a larger number ofK in unknown terrain are larger than those of SNW algorithms with a smaller number ofK (nomatterwhattheteamobjectiveis). Thismeetsourexpectationsincethe response times of the SNW algorithms with a larger number ofK are larger than thoseofSNWalgorithmswithasmallernumberofK inknownterrain. A.3 Complex-TaskAuctionExperiments Weperformthefollowingexperimentstoevaluateourcomplex-taskauctionalgorithms: • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with disjoint coalitions over 100 instances of outdoor-like ter- rainwithrandomlygeneratedlocationsofrobotsandtargets. Wevarythenumber of robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of complex targets from 1, 2, 3, and 4 to 5. We set 323 the robot simple-target capacity to 3. The results are reported in Table A.42 for theteamcosts,andTableA.43fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routingproblemswithdisjointcoalitionsover100instancesofemptyterrainwith randomlygeneratedlocationsofrobotsandtargets. Wevarythenumberofrobots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of complex targets from 1, 2, 3, and 4 to 5. We set the robot simple-target capacity to 3. The results are reported in Table A.44 for the team costs,andTableA.45fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with disjoint coalitions over 100 instances of indoor-like ter- rainwithrandomlygeneratedlocationsofrobotsandtargets. Wevarythenumber of robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of complex targets from 1, 2, 3, and 4 to 5. We set the robot simple-target capacity to 3. The results are reported in Table A.46 for theteamcosts,andTableA.47fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithdifferentrobot simple-target capacities on multi-robot routing problems with disjoint coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robotsandtargets. Wesetthenumberofrobotsto10andthenumberofcomplex targetsto5. Wevarythenumberofsimpletargetsfrom20,30, and40to50, and the robot simple-target capacity from 2, 3, and 4 to 5. The results are reported in TableA.48fortheteamcosts,andTableA.49fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithidealreaction- functioncalculationsonmulti-robotroutingproblemswithdisjointcoalitionsover 324 100instancesofoutdoor-liketerrainwithrandomlygeneratedlocationsofrobots andtargets. Wesetthenumberofrobotsto10andthenumberofcomplextargets to 5. We vary the number of simple targets from 20, 30, and 40 to 50, and the robot simple-target capacity from 2, 3, and 4 to 5. The results are reported in TableA.50fortheteamcosts,andTableA.51fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms with loose robot simple-target capacity constraints on multi-robot routing problems with disjoint coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We set the number of robots to 10, the number ofsimpletargetsto30,andthenumberofcomplextargetsto5. Wevarytherobot simple-target capacity from 3, 4, 5, 6, and 7 to∞. The results are reported in TableA.52fortheteamcosts,andTableA.53fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmsinunknownterrain on multi-robot routing problems with disjoint coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We vary the number of robots from 2, 4, 6, and 8 to 10, the number of simple targets from6,12,18,and24to30,andthenumberofcomplextargetsfrom1,2,3,and 4 to 5. We set the robot simple-target capacity to 3. The results are reported in Table A.54 for the team costs, Table A.55 for the number of repeated executions, andTableA.56forthecumulativeresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlapping coalitions over 100 instances of outdoor-like terrainwithrandomlygeneratedlocationsofrobotsandtargets. Wevarythenum- ber of robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 325 18,and24to30. Wesetthenumberofcomplextargetsto3,andtherobotsimple- target capacity to 3. We set the boundR on the number of related robots to 10 (that is, all robots can be related). The results are reported in Table A.57 for the teamcosts,andTableA.58fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlapping coalitions over 100 instances of empty ter- rainwithrandomlygeneratedlocationsofrobotsandtargets. Wevarythenumber of robots from 2, 4, 6, and 8 to 10 and the number of simple targets from 6, 12, 18,and24to30. Wesetthenumberofcomplextargetsto3,andtherobotsimple- target capacity to 3. We set the boundR on the number of related robots to 10. The results are reported in Table A.59 for the team costs, and Table A.60 for the responsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlapping coalitions over 100 instances of indoor-like terrain with randomly generated locations of robots and targets. We vary the number of robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30. We set the number of complex targets to 3, and the robot simple-target capacity to 3. We set the boundR on the number of related robots to 10. The results are reported in Table A.61 for the team costs, and Table A.62 fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithdifferentrobot simple-target capacities on multi-robot routing problems with overlapping coali- tionsover100instancesofoutdoor-liketerrainwithrandomlygeneratedlocations ofrobotsandtargets. Wesetthenumberofrobotsto10,andthenumberofcom- plextargetsto3. Wevarythenumberofsimpletargetsfrom20,30,and40to50, 326 andtherobotsimple-targetcapacityfrom2,3,and4to5. WesettheboundRon the number of related robots to 10. The results are reported in Table A.63 for the teamcosts,andTableA.64fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithidealreaction- functioncalculationsonmulti-robotroutingproblemswithoverlappingcoalitions over 100 instances of outdoor-like terrain with randomly generated locations of robotsandtargets. Wesetthenumberofrobotsto10andthenumberofcomplex targets to 3. We vary the number of simple targets from 20, 30, and 40 to 50 and thenumberofrobotsimple-targetcapacityfrom2,3,and4to5. Wesetthebound R on the number of related robots to 10. The results are reported in Table A.65 fortheteamcosts,andTableA.66fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms with loose robot simple-target capacity constraints on multi-robot routing problems with overlap- pingcoalitionsover100instancesofoutdoor-liketerrainwithrandomlygenerated locations of robots and targets. We set the number of robots to 10, the number of simple targets to 30, and the number of complex targets to 3. We vary the robot simple-target capacity from 3, 4, 5, 6, and 7 to∞. We set the boundR on the numberofrelatedrobotsto10. TheresultsarereportedinTableA.67fortheteam costs,andTableA.68fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithdifferentnum- bers of complex targets on multi-robot routing problems with overlapping coali- tionsover100instancesofoutdoor-liketerrainwithrandomlygeneratedlocations ofrobotsandtargets. Wesetthenumberofrobotsto10andthenumberofsimple targetsto30. Wevarythenumberofcomplextargetsfrom3,4,5,and6to7. We set the robot simple-target capacity to 3. We set the boundR on the number of 327 relatedrobotsto10. TheresultsarereportedinTableA.69fortheteamcosts,and TableA.70fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmsinunknownterrain on multi-robot routing problems with overlapping coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. Wevarythenumberof robotsfrom 2, 4, 6, and8to 10and thenumberof simple targetsfrom6,12,18,and24to30. Wesetthenumberofcomplextargetsto3,and therobotsimple-targetcapacityto3. WesettheboundRonthenumberofrelated robotsto10. TheresultsarereportedinTableA.71fortheteamcosts,TableA.72 forthenumberofrepeatedexecutions,andTableA.73forthecumulativeresponse times. Wenowexaminetheseresultsinmoredetail. 328 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 246.55 193.76 19.59 100/100 196.81 18.12 87/100 193.76 19.59 100/100 290.04 4 12 2 379.23 313.90 16.08 97/100 317.32 14.55 81/100 313.83 16.04 95/100 581.62 6 18 3 487.97 418.94 13.62 97/100 408.79 15.17 87/100 418.77 13.58 95/100 892.70 8 24 4 593.06 508.34 13.82 98/100 502.03 14.62 94/100 514.52 12.79 94/100 1179.98 10 30 5 660.18 582.47 11.33 93/100 567.32 13.37 89/100 577.35 12.16 94/100 1471.14 MiniMaxTeamObjective 2 6 1 119.50 108.94 8.03 98/100 110.81 6.05 82/100 109.06 7.95 99/100 163.58 4 12 2 115.44 113.47 0.56 68/100 115.70 -2.13 48/100 108.23 5.76 96/100 182.15 6 18 3 114.55 116.63 -3.37 52/100 114.59 -2.60 54/100 104.84 7.68 95/100 194.10 8 24 4 112.99 115.90 -4.28 50/100 112.51 -2.04 49/100 107.12 4.47 85/100 197.27 10 30 5 115.31 119.35 -6.33 46/100 114.58 -2.46 48/100 109.51 3.79 87/100 203.49 MiniLatTeamObjective 2 6 1 357.99 354.68 0.73 97/100 360.52 -1.05 75/100 355.37 0.58 97/100 703.53 4 12 2 565.40 559.94 0.81 68/100 582.03 -3.15 48/100 562.08 0.51 87/100 1413.96 6 18 3 728.57 724.52 0.37 60/100 741.71 -2.31 48/100 723.77 0.63 85/100 2183.76 8 24 4 870.77 864.42 0.67 56/100 890.93 -2.54 38/100 866.46 0.48 79/100 2884.06 10 30 5 1000.40 1001.66 -0.21 48/100 1028.97 -3.09 36/100 996.26 0.41 82/100 3600.49 TableA.42: Complex-TaskAuctions: ComparisonofTeamCostsinOutdoor-LikeTerrain(DisjointCoalitions) Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.00 0.00 0.00 8 24 4 0.00 0.00 0.01 0.01 0.00 10 30 5 0.00 0.00 0.03 0.01 0.00 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.03 0.01 0.00 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.04 0.02 0.00 TableA.43: Complex-TaskAuctions: ComparisonofResponseTimesinOutdoor-LikeTerrain(DisjointCoalitions) 329 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 226.95 178.57 19.38 99/100 179.89 18.66 90/100 178.57 19.38 99/100 279.43 4 12 2 349.18 290.20 15.40 91/100 290.60 14.49 81/100 292.14 14.87 90/100 563.21 6 18 3 437.01 376.85 13.19 97/100 377.66 12.36 83/100 376.96 13.16 97/100 839.71 8 24 4 531.33 462.66 12.23 92/100 451.79 14.03 87/100 457.07 13.29 93/100 1116.10 10 30 5 592.33 527.46 10.32 86/100 511.93 12.76 90/100 523.79 10.94 93/100 1404.23 MiniMaxTeamObjective 2 6 1 107.79 99.51 6.93 97/100 101.55 4.58 82/100 99.63 6.80 98/100 157.10 4 12 2 104.89 103.51 0.37 67/100 103.61 -0.84 55/100 98.99 4.63 90/100 176.42 6 18 3 102.12 102.64 -2.32 59/100 103.72 -4.17 47/100 96.76 3.89 90/100 186.63 8 24 4 101.83 103.35 -2.99 54/100 106.00 -6.43 46/100 95.91 4.92 89/100 191.44 10 30 5 103.45 107.81 -6.94 44/100 104.42 -4.25 54/100 98.49 3.55 85/100 195.53 MiniLatTeamObjective 2 6 1 329.84 326.77 0.73 97/100 330.62 -0.25 83/100 328.44 0.38 98/100 682.61 4 12 2 504.77 504.09 0.12 69/100 519.00 -3.58 41/100 503.75 0.20 83/100 1372.53 6 18 3 649.02 642.93 0.77 62/100 663.29 -2.41 46/100 646.04 0.41 77/100 2050.01 8 24 4 786.35 773.58 1.49 63/100 788.26 -0.63 46/100 780.10 0.74 78/100 2716.89 10 30 5 898.67 895.98 0.09 51/100 913.15 -2.12 46/100 892.21 0.68 79/100 3437.42 TableA.44: Complex-TaskAuctions: ComparisonofTeamCostsinEmptyTerrain(DisjointCoalitions) Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.03 0.01 0.00 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.03 0.01 0.00 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.04 0.02 0.00 TableA.45: Complex-TaskAuctions: ComparisonofResponseTimesinEmptyTerrain(DisjointCoalitions) 330 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 354.07 280.94 18.32 97/100 283.87 17.14 88/100 281.28 18.24 97/100 444.70 4 12 2 543.33 460.72 13.99 96/100 454.16 14.59 85/100 459.74 14.23 98/100 873.03 6 18 3 687.89 591.83 13.06 93/100 593.28 12.43 85/100 589.09 13.37 93/100 1344.70 8 24 4 815.11 711.40 12.04 87/100 702.75 12.96 85/100 709.59 12.31 91/100 1740.61 10 30 5 913.02 818.49 9.66 89/100 804.78 10.90 83/100 807.62 10.88 92/100 2203.08 MiniMaxTeamObjective 2 6 1 167.36 156.69 5.41 96/100 162.16 1.80 75/100 156.33 5.71 99/100 250.35 4 12 2 162.42 160.89 0.06 73/100 162.62 -1.58 52/100 155.57 3.87 93/100 271.40 6 18 3 160.26 160.12 -0.74 64/100 162.20 -3.11 50/100 154.02 3.60 92/100 292.36 8 24 4 158.44 164.84 -5.77 51/100 165.32 -6.54 40/100 150.58 4.36 89/100 291.69 10 30 5 160.89 166.12 -4.79 44/100 164.31 -3.63 46/100 153.46 3.85 80/100 304.48 MiniLatTeamObjective 2 6 1 521.40 519.34 0.28 90/100 524.75 -0.88 80/100 518.13 0.47 93/100 1100.24 4 12 2 797.16 787.82 1.00 69/100 811.71 -1.99 51/100 794.25 0.32 87/100 2155.33 6 18 3 1016.89 998.47 1.41 74/100 1019.55 -1.03 47/100 1006.67 0.82 85/100 3292.32 8 24 4 1203.52 1194.10 0.41 59/100 1221.39 -2.01 45/100 1195.42 0.58 82/100 4240.61 10 30 5 1366.16 1362.57 0.12 48/100 1377.95 -1.30 49/100 1365.34 0.13 71/100 5402.24 TableA.46: Complex-TaskAuctions: ComparisonofTeamCostsinIndoor-LikeTerrain(DisjointCoalitions) Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.01 0.01 0.00 10 30 5 0.00 0.00 0.03 0.01 0.00 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.03 0.01 0.00 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.00 0.00 0.00 6 18 3 0.00 0.00 0.01 0.00 0.00 8 24 4 0.00 0.00 0.02 0.01 0.00 10 30 5 0.00 0.00 0.04 0.02 0.00 TableA.47: Complex-TaskAuctions: ComparisonofResponseTimesinIndoor-LikeTerrain(DisjointCoalitions) 331 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex Simple-Target SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets Capacities (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 5 2 574.30 519.46 9.24 87/100 503.72 11.67 86/100 517.68 9.56 92/100 1142.38 10 30 5 3 660.18 582.47 11.33 93/100 567.32 13.37 89/100 577.35 12.16 94/100 1471.14 10 40 5 4 725.13 628.77 12.81 94/100 615.04 14.42 93/100 629.27 12.71 98/100 1833.78 10 50 5 5 796.32 681.08 14.22 97/100 665.88 16.04 98/100 673.05 15.21 98/100 2165.77 MiniMaxTeamObjective 10 20 5 2 99.70 109.40 -12.31 38/100 103.52 -6.31 44/100 97.34 2.04 89/100 164.64 10 30 5 3 115.31 119.35 -6.33 46/100 114.58 -2.46 48/100 109.51 3.79 87/100 203.49 10 40 5 4 124.67 125.67 -2.35 49/100 121.92 0.48 56/100 117.03 5.25 73/100 246.18 10 50 5 5 131.63 133.66 -3.46 42/100 129.34 -0.10 56/100 119.54 7.99 77/100 287.25 MiniLatTeamObjective 10 20 5 2 687.16 689.95 -2.33 52/100 700.96 -2.38 37/100 685.25 0.27 94/100 2162.04 10 30 5 3 1000.40 1001.66 -0.21 48/100 1028.97 -3.09 36/100 996.26 0.41 82/100 3600.49 10 40 5 4 1353.17 1336.55 1.18 60/100 1368.34 -1.33 45/100 1341.16 0.88 69/100 5480.09 10 50 5 5 1735.43 1699.35 1.99 68/100 1746.27 -0.97 47/100 1708.96 1.45 64/100 7598.29 Table A.48: Complex-Task Auctions: Comparison of Team Costs for Different Robot Simple-Target Capacities (Disjoint Coalitions) Robots SimpleTargets ComplexTargets Simple-TargetCapacities SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 20 5 2 0.00 0.00 0.00 0.00 0.00 10 30 5 3 0.00 0.00 0.03 0.01 0.00 10 40 5 4 0.00 0.03 0.10 0.10 0.00 10 50 5 5 0.00 0.09 0.51 0.37 0.00 MiniMaxTeamObjective 10 20 5 2 0.00 0.00 0.00 0.00 0.00 10 30 5 3 0.00 0.00 0.03 0.01 0.00 10 40 5 4 0.00 0.04 0.09 0.08 0.00 10 50 5 5 0.00 0.11 0.60 0.35 0.00 MiniLatTeamObjective 10 20 5 2 0.00 0.00 0.01 0.00 0.00 10 30 5 3 0.00 0.00 0.04 0.02 0.00 10 40 5 4 0.00 0.04 0.18 0.11 0.00 10 50 5 5 0.00 0.12 0.98 0.47 0.00 Table A.49: Complex-Task Auctions: Comparison of Response Times for Different Robot Simple-Target Capacities (Disjoint Coalitions) 332 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex Simple-Target SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets Capacities (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 20 5 2 574.30 519.42 9.25 87/100 503.23 11.76 86/100 517.62 9.57 92/100 1142.38 3 30 5 3 660.18 581.05 11.53 93/100 565.12 13.68 89/100 577.11 12.20 94/100 1471.14 4 40 5 4 725.13 629.13 12.76 93/100 613.19 14.69 94/100 627.80 12.93 98/100 1833.78 5 50 5 5 796.32 673.95 15.10 98/100 659.72 16.83 97/100 668.32 15.81 98/100 2165.77 MiniMaxTeamObjective 2 20 5 2 99.70 109.33 -12.24 38/100 103.49 -6.30 44/100 97.33 2.06 89/100 164.64 3 30 5 3 115.31 119.21 -6.21 46/100 114.34 -2.20 48/100 109.51 3.79 87/100 203.49 4 40 5 4 124.67 125.48 -2.20 49/100 121.49 0.90 57/100 117.36 4.99 73/100 246.18 5 50 5 5 131.63 130.89 -1.33 50/100 125.51 2.90 59/100 118.88 8.44 79/100 287.25 MiniLatTeamObjective 2 20 5 2 687.16 688.39 -0.52 50/100 700.62 -2.36 37/100 684.80 0.33 99/100 2162.04 3 30 5 3 1000.40 999.59 0.02 47/100 1025.39 -2.74 38/100 993.99 0.63 95/100 3600.49 4 40 5 4 1353.17 1331.78 1.52 64/100 1357.46 -0.55 50/100 1334.43 1.37 85/100 5480.09 5 50 5 5 1735.43 1692.61 2.33 72/100 1746.77 -1.01 44/100 1703.08 1.78 77/100 7598.29 Table A.50: Complex-Task Auctions: Comparison of Team Costs for Ideal Reaction-Function Calculations (Disjoint Coali- tions) Robots SimpleTargets ComplexTargets Simple-TargetCapacities SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 20 5 2 0.00 0.00 0.00 0.00 0.00 10 30 5 3 0.00 0.01 0.04 0.02 0.00 10 40 5 4 0.00 0.04 0.12 0.11 0.00 10 50 5 5 0.00 0.12 0.61 0.44 0.00 MiniMaxTeamObjective 10 20 5 2 0.00 0.00 0.00 0.00 0.00 10 30 5 3 0.00 0.01 0.04 0.02 0.00 10 40 5 4 0.00 0.05 0.12 0.10 0.00 10 50 5 5 0.00 0.14 0.72 0.41 0.00 MiniLatTeamObjective 10 20 5 2 0.00 0.00 0.00 0.00 0.00 10 30 5 3 0.00 0.01 0.05 0.03 0.00 10 40 5 4 0.00 0.05 0.22 0.13 0.00 10 50 5 5 0.00 0.15 1.15 0.57 0.00 Table A.51: Complex-Task Auctions: Comparison of Response Times for Ideal Reaction-Function Calculations (Disjoint Coalitions) 333 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex Simple-Target SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets Capacities (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 3 30 5 3 660.18 582.47 11.33 93/100 567.32 13.37 89/100 577.35 12.16 94/100 1471.14 4 30 5 4 576.31 514.40 10.35 89/100 485.45 15.13 95/100 515.18 10.21 90/100 1472.93 5 30 5 5 559.93 494.63 11.26 93/100 458.16 17.64 98/100 496.52 11.05 93/100 1462.98 6 30 5 6 552.73 485.20 11.69 87/100 444.64 18.96 98/100 485.17 11.71 86/100 1462.98 7 30 5 7 538.71 481.74 10.23 88/100 438.46 18.20 99/100 482.01 10.19 88/100 1462.98 30 30 5 ∞ 549.75 475.36 13.02 93/100 427.43 21.75 100/100 475.00 13.09 93/100 1462.98 MiniMaxTeamObjective 3 30 5 3 115.31 119.35 -6.33 46/100 114.58 -2.46 48/100 109.51 3.79 87/100 203.49 4 30 5 4 94.18 94.63 -3.03 56/100 75.57 16.88 84/100 90.48 2.43 80/100 230.26 5 30 5 5 94.06 93.46 -1.44 65/100 70.36 22.98 94/100 89.72 3.60 81/100 248.89 6 30 5 6 94.54 92.17 0.03 64/100 68.08 25.54 94/100 89.76 4.01 78/100 248.89 7 30 5 7 94.77 91.73 0.82 66/100 67.59 26.23 93/100 89.83 4.04 75/100 248.89 30 30 5 ∞ 95.16 91.45 1.61 69/100 67.49 26.76 94/100 89.54 4.87 77/100 248.89 MiniLatTeamObjective 3 30 5 3 1000.40 1001.66 -0.21 48/100 1028.97 -3.09 36/100 996.26 0.41 82/100 3600.49 4 30 5 4 871.95 865.89 0.55 56/100 894.77 -2.87 34/100 865.82 0.66 89/100 3885.91 5 30 5 5 850.50 840.80 1.03 69/100 869.64 -2.42 36/100 847.45 0.33 79/100 4002.94 6 30 5 6 844.04 836.95 0.73 67/100 862.57 -2.41 35/100 842.11 0.20 73/100 4002.94 7 30 5 7 842.67 835.13 0.78 67/100 860.38 -2.32 35/100 840.00 0.26 75/100 4002.94 30 30 5 ∞ 842.80 834.33 0.88 69/100 859.92 -2.26 36/100 839.61 0.32 78/100 4002.94 TableA.52: Complex-TaskAuctions: ComparisonofTeamCostsforLooseRobotSimple-TargetCapacityConstraints(Disjoint Coalitions) Robots SimpleTargets ComplexTargets Simple-TargetCapacities SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 30 5 3 0.00 0.00 0.03 0.01 0.00 10 30 5 4 0.00 0.02 0.07 0.05 0.00 10 30 5 5 0.00 0.03 0.14 0.14 0.00 10 30 5 6 0.00 0.08 0.32 0.36 0.00 10 30 5 7 0.00 0.13 0.60 0.54 0.00 10 30 5 ∞ 0.00 0.95 2.24 2.00 0.00 MiniMaxTeamObjective 10 30 5 3 0.00 0.00 0.03 0.01 0.00 10 30 5 4 0.00 0.02 0.08 0.04 0.00 10 30 5 5 0.00 0.03 0.21 0.06 0.00 10 30 5 6 0.00 0.04 0.39 0.09 0.00 10 30 5 7 0.00 0.05 0.49 0.10 0.00 10 30 5 ∞ 0.00 0.05 0.60 0.11 0.00 MiniLatTeamObjective 10 30 5 3 0.00 0.00 0.04 0.02 0.00 10 30 5 4 0.00 0.02 0.05 0.04 0.00 10 30 5 5 0.00 0.03 0.17 0.08 0.00 10 30 5 6 0.00 0.05 0.37 0.12 0.00 10 30 5 7 0.00 0.15 0.54 0.16 0.00 10 30 5 ∞ 0.00 0.16 0.85 0.20 0.00 Table A.53: Complex-Task Auctions: Comparison of Response Times for Loose Robot Simple-Target Capacity Constraints (DisjointCoalitions) 334 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 304.38 254.52 14.00 77/100 255.58 13.41 76/100 255.41 13.67 77/100 1087.37 4 12 2 463.84 408.88 10.71 82/100 404.91 10.79 72/100 402.79 11.98 85/100 3065.27 6 18 3 588.29 540.23 7.28 74/100 538.44 7.14 69/100 533.97 8.24 73/100 5230.18 8 24 4 694.05 664.50 3.44 66/100 647.06 5.44 63/100 642.08 6.51 66/100 7155.15 10 30 5 767.94 732.98 3.88 63/100 726.14 4.76 63/100 726.20 4.98 71/100 9646.11 MiniMaxTeamObjective 2 6 1 151.25 142.23 4.33 73/100 149.56 -0.72 62/100 142.90 3.93 72/100 650.79 4 12 2 141.02 154.54 -11.74 40/100 151.55 -9.82 45/100 146.87 -5.48 42/100 1002.26 6 18 3 146.71 156.89 -9.40 37/100 154.01 -9.07 45/100 146.21 -1.80 53/100 1257.41 8 24 4 137.17 150.50 -11.73 35/100 156.27 -16.94 30/100 141.66 -4.77 51/100 1338.25 10 30 5 136.78 160.72 -20.33 25/100 148.01 -11.64 39/100 142.16 -6.85 54/100 1455.60 MiniLatTeamObjective 2 6 1 459.14 451.26 1.06 81/100 446.95 1.86 75/100 451.36 1.04 81/100 2450.61 4 12 2 695.67 698.70 -0.85 53/100 705.34 -1.52 47/100 687.19 1.15 78/100 6991.06 6 18 3 906.70 892.09 1.32 64/100 919.90 -1.97 52/100 885.46 2.06 78/100 12777.91 8 24 4 1063.16 1059.73 0.00 53/100 1073.10 -1.42 50/100 1059.44 0.17 67/100 17945.39 10 30 5 1215.71 1203.73 0.70 54/100 1229.62 -1.73 44/100 1213.86 -0.12 58/100 24981.61 TableA.54: Complex-TaskAuctions: ComparisonofTeamCostsinUnknownTerrain(DisjointCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 80.82 76.33 3.69 64/100 78.27 0.97 53/100 76.48 3.54 62/100 128.68 4 12 2 96.63 93.76 2.09 59/100 94.08 0.80 55/100 92.70 3.07 66/100 208.44 6 18 3 97.75 95.01 1.96 59/100 101.44 -5.49 45/100 95.26 1.62 55/100 242.06 8 24 4 96.04 98.06 -3.43 52/100 100.23 -6.56 44/100 97.18 -2.82 47/100 253.79 10 30 5 94.77 94.32 -0.51 52/100 97.11 -3.70 43/100 96.57 -2.79 53/100 264.29 MiniMaxTeamObjective 2 6 1 78.75 74.91 3.49 74/100 77.29 0.06 62/100 74.73 3.84 74/100 128.68 4 12 2 92.97 97.37 -5.70 37/100 96.39 -4.92 44/100 95.34 -3.42 49/100 208.44 6 18 3 97.71 101.35 -4.59 40/100 100.95 -4.92 41/100 98.50 -1.67 53/100 242.06 8 24 4 94.51 101.85 -8.83 29/100 103.12 -10.79 33/100 96.67 -3.02 54/100 253.79 10 30 5 94.98 104.56 -11.36 25/100 101.45 -8.23 37/100 97.08 -3.50 47/100 264.29 MiniLatTeamObjective 2 6 1 78.75 78.14 0.13 82/100 78.17 0.19 70/100 78.15 0.14 81/100 128.68 4 12 2 92.97 95.04 -3.09 47/100 91.53 0.71 62/100 91.94 0.67 71/100 208.44 6 18 3 97.71 97.82 -0.75 56/100 97.23 -0.30 58/100 96.83 0.63 69/100 242.06 8 24 4 94.51 97.85 -4.06 42/100 95.50 -2.68 48/100 95.26 -1.26 60/100 253.79 10 30 5 94.98 97.15 -3.18 38/100 94.97 -1.38 56/100 96.22 -1.78 55/100 264.29 Table A.55: Complex-Task Auctions: Comparison of Numbers of Repeated Executions in Unknown Terrain (Disjoint Coali- tions) 335 Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 1 0.04 0.05 0.07 0.05 0.06 4 12 2 0.09 0.13 0.24 0.17 0.21 6 18 3 0.14 0.23 0.55 0.36 0.38 8 24 4 0.18 0.37 0.97 0.59 0.55 10 30 5 0.23 0.51 1.46 0.89 0.73 MiniMaxTeamObjective 2 6 1 0.04 0.04 0.07 0.05 0.06 4 12 2 0.09 0.13 0.29 0.17 0.21 6 18 3 0.14 0.25 0.64 0.34 0.38 8 24 4 0.19 0.40 1.15 0.56 0.55 10 30 5 0.25 0.60 1.77 0.83 0.73 MiniLatTeamObjective 2 6 1 0.04 0.04 0.07 0.05 0.06 4 12 2 0.09 0.13 0.26 0.17 0.21 6 18 3 0.15 0.22 0.56 0.33 0.38 8 24 4 0.19 0.35 1.02 0.55 0.55 10 30 5 0.24 0.50 1.61 0.79 0.72 TableA.56: Complex-TaskAuctions: ComparisonofCumulativeResponseTimesinUnknownTerrain(DisjointCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 342.20 274.86 18.83 99/100 276.30 18.17 94/100 274.92 18.81 99/100 431.78 4 12 3 372.05 326.45 11.79 95/100 326.80 11.42 88/100 329.17 11.04 94/100 651.99 6 18 3 403.32 364.07 9.43 89/100 374.48 6.40 70/100 368.61 8.22 90/100 971.22 8 24 3 434.54 396.71 8.51 95/100 405.65 6.13 70/100 404.97 6.69 92/100 1084.38 10 30 3 474.34 439.28 7.26 92/100 436.14 7.77 85/100 446.55 5.79 86/100 1340.91 MiniMaxTeamObjective 2 6 3 158.21 143.91 8.35 94/100 147.29 5.81 75/100 143.75 8.47 95/100 230.87 4 12 3 120.22 106.78 9.40 78/100 122.46 -4.70 42/100 113.43 4.52 77/100 202.24 6 18 3 110.90 94.67 12.55 83/100 113.45 -5.42 49/100 104.26 4.64 74/100 236.80 8 24 3 105.26 91.51 10.48 85/100 102.95 -1.93 57/100 100.43 3.71 80/100 190.51 10 30 3 101.39 88.81 10.20 86/100 101.66 -2.57 46/100 97.43 3.06 82/100 217.21 MiniLatTeamObjective 2 6 3 616.69 598.69 2.57 93/100 613.53 -0.24 64/100 600.77 2.25 93/100 1313.20 4 12 3 650.21 628.57 2.93 81/100 681.65 -5.40 38/100 640.52 1.35 88/100 1695.85 6 18 3 716.98 684.67 4.25 85/100 743.97 -4.10 45/100 710.31 0.95 80/100 2518.37 8 24 3 775.47 747.87 2.94 73/100 793.04 -3.33 38/100 774.49 -0.18 71/100 2563.85 10 30 3 855.40 817.62 4.09 88/100 871.86 -2.35 45/100 848.30 0.79 88/100 3142.19 TableA.57: Complex-TaskAuctions: ComparisonofTeamCostsinOutdoor-LikeTerrain(OverlappingCoalitions) 336 Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 3 0.00 0.01 0.03 0.01 0.00 4 12 3 0.00 0.01 0.02 0.01 0.00 6 18 3 0.00 0.01 0.01 0.01 0.00 8 24 3 0.00 0.01 0.01 0.01 0.00 10 30 3 0.00 0.01 0.01 0.01 0.00 MiniMaxTeamObjective 2 6 3 0.00 0.01 0.03 0.01 0.00 4 12 3 0.00 0.01 0.01 0.01 0.00 6 18 3 0.00 0.01 0.01 0.01 0.00 8 24 3 0.00 0.01 0.01 0.01 0.00 10 30 3 0.00 0.01 0.01 0.01 0.00 MiniLatTeamObjective 2 6 3 0.00 0.01 0.06 0.03 0.00 4 12 3 0.00 0.01 0.04 0.02 0.00 6 18 3 0.00 0.01 0.02 0.02 0.00 8 24 3 0.00 0.01 0.02 0.01 0.00 10 30 3 0.00 0.01 0.02 0.01 0.00 TableA.58: Complex-TaskAuctions: ComparisonofResponseTimesinOutdoor-LikeTerrain(OverlappingCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 310.70 246.90 19.18 98/100 252.37 17.15 92/100 246.98 19.15 98/100 408.05 4 12 3 330.80 290.47 11.64 93/100 297.48 9.32 77/100 295.24 10.25 92/100 632.47 6 18 3 358.91 323.98 9.31 92/100 328.86 7.63 78/100 327.37 8.41 90/100 917.65 8 24 3 392.98 358.47 8.51 95/100 365.80 6.26 75/100 364.32 7.09 90/100 1040.07 10 30 3 419.66 389.25 7.03 94/100 391.99 6.15 69/100 398.26 4.95 85/100 1246.30 MiniMaxTeamObjective 2 6 3 144.96 129.84 9.07 89/100 135.76 4.64 73/100 129.95 9.04 91/100 219.52 4 12 3 107.12 95.02 9.50 79/100 111.32 -6.82 46/100 100.23 5.24 81/100 197.95 6 18 3 96.90 84.28 10.69 83/100 102.62 -10.56 46/100 94.23 1.05 70/100 226.52 8 24 3 93.58 81.71 10.79 85/100 96.43 -6.85 48/100 89.99 2.69 78/100 184.95 10 30 3 91.07 81.66 8.17 82/100 92.09 -6.00 54/100 88.35 1.82 77/100 191.41 MiniLatTeamObjective 2 6 3 562.87 545.08 2.86 93/100 563.41 -0.68 65/100 545.97 2.71 93/100 1252.47 4 12 3 568.54 555.06 2.11 76/100 603.99 -6.88 38/100 565.05 0.54 81/100 1657.58 6 18 3 630.03 611.08 2.78 76/100 657.52 -4.76 35/100 627.39 0.28 75/100 2379.20 8 24 3 700.42 670.50 3.83 80/100 712.08 -2.15 45/100 695.51 0.58 86/100 2455.84 10 30 3 761.24 731.85 3.54 86/100 774.52 -2.41 40/100 757.80 0.41 80/100 2912.92 TableA.59: Complex-TaskAuctions: ComparisonofTeamCostsinEmptyTerrain(OverlappingCoalitions) 337 Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 3 0.00 0.01 0.03 0.01 0.00 4 12 3 0.00 0.01 0.02 0.01 0.00 6 18 3 0.00 0.01 0.01 0.01 0.00 8 24 3 0.00 0.01 0.01 0.01 0.00 10 30 3 0.00 0.01 0.01 0.01 0.00 MiniMaxTeamObjective 2 6 3 0.00 0.01 0.03 0.01 0.00 4 12 3 0.00 0.01 0.01 0.01 0.00 6 18 3 0.00 0.01 0.01 0.01 0.00 8 24 3 0.00 0.01 0.01 0.01 0.00 10 30 3 0.00 0.01 0.01 0.01 0.00 MiniLatTeamObjective 2 6 3 0.00 0.01 0.06 0.03 0.00 4 12 3 0.00 0.01 0.04 0.02 0.00 6 18 3 0.00 0.01 0.02 0.02 0.00 8 24 3 0.00 0.01 0.02 0.01 0.00 10 30 3 0.00 0.01 0.02 0.01 0.00 TableA.60: Complex-TaskAuctions: ComparisonofResponseTimesinEmptyTerrain(OverlappingCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 500.23 404.96 18.10 98/100 408.62 17.31 95/100 405.38 18.03 98/100 658.67 4 12 3 521.75 464.79 10.31 95/100 469.90 9.01 85/100 469.18 9.55 92/100 983.69 6 18 3 569.64 509.95 10.03 90/100 522.20 7.73 83/100 516.17 9.02 89/100 1481.25 8 24 3 604.52 550.40 8.65 97/100 564.36 6.10 76/100 558.38 7.42 92/100 1632.44 10 30 3 646.68 599.44 7.03 91/100 602.15 6.49 71/100 607.38 5.85 87/100 1967.98 MiniMaxTeamObjective 2 6 3 231.86 213.04 7.24 86/100 218.22 4.84 71/100 212.77 7.36 86/100 353.78 4 12 3 168.52 155.66 6.07 75/100 181.92 -10.46 39/100 163.01 2.54 72/100 308.21 6 18 3 149.73 131.52 10.82 84/100 166.23 -14.85 39/100 141.44 4.25 77/100 356.56 8 24 3 139.82 125.51 7.66 76/100 154.87 -16.97 35/100 140.62 -1.97 68/100 286.08 10 30 3 143.43 127.83 9.48 86/100 146.02 -5.12 50/100 137.92 2.97 82/100 315.49 MiniLatTeamObjective 2 6 3 916.14 892.35 2.20 87/100 903.77 0.69 69/100 892.33 2.23 91/100 2026.69 4 12 3 913.17 883.05 2.93 77/100 949.87 -4.64 41/100 907.20 0.57 82/100 2591.70 6 18 3 986.83 949.41 3.43 81/100 1004.23 -2.47 48/100 977.30 0.87 78/100 3867.27 8 24 3 1057.02 1020.13 3.13 70/100 1092.39 -3.81 36/100 1050.47 0.56 77/100 3855.45 10 30 3 1155.21 1114.68 3.29 76/100 1164.23 -1.17 47/100 1151.24 0.29 74/100 4650.90 TableA.61: Complex-TaskAuctions: ComparisonofTeamCostsinIndoor-LikeTerrain(OverlappingCoalitions) 338 Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 3 0.00 0.01 0.03 0.01 0.00 4 12 3 0.00 0.01 0.02 0.01 0.00 6 18 3 0.00 0.01 0.01 0.01 0.00 8 24 3 0.00 0.01 0.01 0.01 0.00 10 30 3 0.00 0.01 0.01 0.01 0.00 MiniMaxTeamObjective 2 6 3 0.00 0.01 0.03 0.01 0.00 4 12 3 0.00 0.01 0.01 0.01 0.00 6 18 3 0.00 0.01 0.01 0.01 0.00 8 24 3 0.00 0.01 0.01 0.01 0.00 10 30 3 0.00 0.01 0.01 0.01 0.00 MiniLatTeamObjective 2 6 3 0.00 0.01 0.06 0.03 0.00 4 12 3 0.00 0.01 0.04 0.02 0.00 6 18 3 0.00 0.01 0.02 0.02 0.00 8 24 3 0.00 0.01 0.02 0.01 0.00 10 30 3 0.00 0.01 0.02 0.01 0.00 TableA.62: Complex-TaskAuctions: ComparisonofResponseTimesinIndoor-LikeTerrain(OverlappingCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex Simple-Target SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets Capacities (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 3 2 397.18 365.50 7.82 96/100 375.35 5.19 70/100 373.86 5.87 89/100 975.86 10 30 3 3 474.34 439.28 7.26 92/100 436.14 7.77 85/100 446.55 5.79 86/100 1340.91 10 40 3 4 532.16 486.55 8.36 97/100 497.67 5.94 75/100 494.36 6.91 92/100 1692.78 10 50 3 5 582.91 533.31 8.38 97/100 549.83 5.44 79/100 542.14 6.88 94/100 2008.83 MiniMaxTeamObjective 10 20 3 2 84.04 72.65 11.41 87/100 92.70 -16.34 45/100 82.06 0.86 84/100 162.83 10 30 3 3 101.39 88.81 10.20 86/100 101.66 -2.57 46/100 97.43 3.06 82/100 217.21 10 40 3 4 110.81 94.46 12.32 82/100 114.63 -7.07 48/100 107.87 0.75 69/100 253.02 10 50 3 5 121.10 106.32 9.97 81/100 121.68 -4.12 52/100 111.41 6.48 75/100 287.84 MiniLatTeamObjective 10 20 3 2 556.51 532.74 3.86 78/100 574.77 -3.61 34/100 555.10 0.17 89/100 1774.52 10 30 3 3 855.40 817.62 4.09 88/100 871.86 -2.35 45/100 848.30 0.79 88/100 3142.19 10 40 3 4 1187.15 1132.76 4.27 88/100 1215.19 -2.82 43/100 1177.79 0.73 80/100 4832.00 10 50 3 5 1565.89 1502.56 3.75 79/100 1574.46 -0.86 47/100 1547.15 1.05 64/100 6790.31 Table A.63: Complex-Task Auctions: Comparison of Team Costs for Different Robot Simple-Target Capacities (Overlapping Coalitions) 339 Robots SimpleTargets ComplexTargets Simple-TargetCapacities SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 20 3 2 0.00 0.00 0.00 0.00 0.00 10 30 3 3 0.00 0.01 0.01 0.01 0.00 10 40 3 4 0.01 0.02 0.02 0.01 0.00 10 50 3 5 0.01 0.04 0.06 0.04 0.00 MiniMaxTeamObjective 10 20 3 2 0.00 0.00 0.00 0.00 0.00 10 30 3 3 0.00 0.01 0.01 0.01 0.00 10 40 3 4 0.00 0.03 0.02 0.01 0.00 10 50 3 5 0.01 0.04 0.05 0.04 0.00 MiniLatTeamObjective 10 20 3 2 0.00 0.00 0.00 0.00 0.00 10 30 3 3 0.00 0.01 0.02 0.01 0.00 10 40 3 4 0.01 0.02 0.07 0.02 0.00 10 50 3 5 0.01 0.08 0.20 0.05 0.00 TableA.64: Complex-TaskAuctions: ComparisonofResponseTimesforDifferentRobotSimple-TargetCapacities(Overlap- pingCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex Simple-Target SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets Capacities (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 3 2 397.18 364.82 8.00 96/100 375.17 5.24 69/100 373.25 6.03 90/100 975.86 10 30 3 3 474.34 438.19 7.49 92/100 434.89 8.03 86/100 445.98 5.92 87/100 1340.91 10 40 3 4 532.16 484.73 8.69 98/100 494.55 6.56 78/100 493.00 7.17 95/100 1692.78 10 50 3 5 582.91 530.16 8.93 99/100 547.25 5.90 80/100 540.58 7.17 94/100 2008.83 MiniMaxTeamObjective 10 20 3 2 84.04 72.64 11.43 87/100 91.87 -15.11 44/100 81.62 1.42 85/100 162.83 10 30 3 3 101.39 88.77 10.23 86/100 100.79 -1.89 47/100 97.45 2.94 82/100 217.21 10 40 3 4 110.81 94.36 12.42 82/100 113.97 -6.46 47/100 107.33 1.05 70/100 253.02 10 50 3 5 121.10 106.32 9.97 81/100 121.83 -4.14 53/100 110.78 6.95 79/100 287.84 MiniLatTeamObjective 10 20 3 2 556.51 532.57 3.89 79/100 573.66 -3.42 35/100 554.31 0.30 94/100 1774.52 10 30 3 3 855.40 816.86 4.19 89/100 868.98 -1.97 46/100 848.59 0.77 93/100 3142.19 10 40 3 4 1187.15 1131.26 4.40 88/100 1207.25 -2.20 45/100 1173.66 1.07 90/100 4832.00 10 50 3 5 1565.89 1496.34 4.15 83/100 1560.28 -0.02 53/100 1534.65 1.86 84/100 6790.31 TableA.65: Complex-TaskAuctions: ComparisonofTeamCostsforIdealReaction-FunctionCalculations(OverlappingCoali- tions) 340 Robots SimpleTargets ComplexTargets Simple-TargetCapacities SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 20 3 2 0.00 0.00 0.00 0.00 0.00 10 30 3 3 0.00 0.01 0.01 0.01 0.00 10 40 3 4 0.01 0.02 0.03 0.01 0.00 10 50 3 5 0.01 0.05 0.12 0.05 0.00 MiniMaxTeamObjective 10 20 3 2 0.00 0.00 0.00 0.00 0.00 10 30 3 3 0.00 0.00 0.01 0.01 0.00 10 40 3 4 0.00 0.01 0.04 0.01 0.00 10 50 3 5 0.01 0.08 0.10 0.05 0.00 MiniLatTeamObjective 10 20 3 2 0.00 0.00 0.00 0.00 0.00 10 30 3 3 0.00 0.01 0.03 0.02 0.00 10 40 3 4 0.01 0.02 0.12 0.03 0.00 10 50 3 5 0.01 0.12 0.34 0.08 0.00 Table A.66: Complex-Task Auctions: Comparison of Response Times for Ideal Reaction-Function Calculations (Overlapping Coalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex Simple-Target SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets Capacities (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 30 3 3 474.34 439.28 7.26 92/100 436.14 7.77 85/100 446.55 5.79 86/100 1340.91 10 30 3 4 394.38 360.92 8.29 93/100 363.11 7.47 82/100 363.49 7.57 92/100 1301.69 10 30 3 5 369.09 338.69 8.02 90/100 338.74 7.77 83/100 338.44 8.10 89/100 1316.08 10 30 3 6 358.76 329.64 7.97 93/100 327.66 8.44 87/100 328.50 8.28 94/100 1316.08 10 30 3 7 354.08 324.98 8.09 96/100 320.68 9.17 91/100 324.97 8.08 96/100 1316.08 10 30 3 ∞ 345.48 315.24 8.63 95/100 311.48 9.69 95/100 314.90 8.72 96/100 1316.08 MiniMaxTeamObjective 10 30 3 3 101.39 88.81 10.20 86/100 101.66 -2.57 46/100 97.43 3.06 82/100 217.21 10 30 3 4 63.67 58.75 6.63 80/100 65.03 -4.54 49/100 62.33 1.68 82/100 229.03 10 30 3 5 58.25 55.25 4.11 72/100 59.07 -3.29 53/100 58.04 0.32 75/100 268.10 10 30 3 6 56.87 54.89 2.67 71/100 57.23 -2.11 55/100 56.14 0.80 76/100 268.10 10 30 3 7 56.21 54.72 1.90 69/100 56.82 -2.45 56/100 55.78 0.26 75/100 268.10 10 30 3 ∞ 56.02 54.67 1.73 69/100 56.73 -2.55 56/100 55.51 0.41 76/100 268.10 MiniLatTeamObjective 10 30 3 3 855.40 817.62 4.09 88/100 871.86 -2.35 45/100 848.30 0.79 88/100 3142.19 10 30 3 4 720.17 707.77 1.61 76/100 735.30 -2.46 38/100 716.46 0.50 83/100 3438.95 10 30 3 5 695.89 688.16 1.00 69/100 713.22 -2.82 37/100 693.69 0.29 80/100 3647.22 10 30 3 6 689.82 683.82 0.78 69/100 707.02 -2.77 37/100 687.59 0.31 83/100 3647.22 10 30 3 7 688.35 682.70 0.73 69/100 705.36 -2.74 37/100 686.39 0.27 82/100 3647.22 10 30 3 ∞ 688.14 682.63 0.71 69/100 705.15 -2.74 36/100 686.12 0.28 83/100 3647.22 TableA.67: Complex-TaskAuctions: Comparisonof TeamCosts for Loose RobotSimple-TargetCapacity Constraints (Over- lappingCoalitions) 341 Robots SimpleTargets ComplexTargets Simple-TargetCapacities SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 30 3 3 0.00 0.01 0.01 0.01 0.00 10 30 3 4 0.00 0.01 0.08 0.01 0.00 10 30 3 5 0.00 0.02 0.17 0.02 0.00 10 30 3 6 0.00 0.04 0.30 0.04 0.00 10 30 3 7 0.00 0.06 0.42 0.06 0.00 10 30 3 ∞ 0.00 0.32 0.80 0.33 0.00 MiniMaxTeamObjective 10 30 3 3 0.00 0.01 0.01 0.01 0.00 10 30 3 4 0.00 0.01 0.01 0.01 0.00 10 30 3 5 0.00 0.01 0.01 0.01 0.00 10 30 3 6 0.00 0.01 0.01 0.01 0.00 10 30 3 7 0.00 0.01 0.01 0.01 0.00 10 30 3 ∞ 0.00 0.01 0.01 0.01 0.00 MiniLatTeamObjective 10 30 3 3 0.00 0.01 0.02 0.01 0.00 10 30 3 4 0.00 0.01 0.01 0.01 0.00 10 30 3 5 0.00 0.01 0.01 0.01 0.00 10 30 3 6 0.00 0.01 0.01 0.01 0.00 10 30 3 7 0.00 0.01 0.01 0.01 0.00 10 30 3 ∞ 0.00 0.01 0.01 0.01 0.00 Table A.68: Complex-Task Auctions: Comparison of Response Times for Loose Robot Simple-Target Capacity Constraints (OverlappingCoalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 30 3 474.34 439.28 7.26 92/100 436.14 7.77 85/100 446.55 5.79 86/100 1340.91 10 30 4 504.24 459.84 7.76 74/100 460.64 7.66 72/100 471.56 5.40 72/100 1395.93 10 30 5 530.83 481.22 8.50 76/100 486.48 7.52 70/100 491.10 6.72 72/100 1539.48 10 30 6 560.95 510.99 8.08 74/100 511.07 7.83 71/100 520.74 6.22 66/100 1716.98 10 30 7 595.51 542.36 8.12 75/100 537.66 9.07 79/100 557.94 5.49 70/100 1895.78 MiniMaxTeamObjective 10 30 3 101.39 88.81 10.20 86/100 101.66 -2.57 46/100 97.43 3.06 82/100 217.21 10 30 4 103.46 89.54 9.43 68/100 111.42 -12.87 47/100 100.60 -1.51 52/100 217.69 10 30 5 108.13 88.41 15.67 76/100 114.18 -9.58 39/100 101.08 3.47 64/100 244.05 10 30 6 114.17 90.04 17.81 86/100 104.43 4.58 58/100 110.95 -1.08 57/100 279.80 10 30 7 113.42 95.19 13.04 76/100 116.63 -7.09 46/100 107.16 1.69 59/100 314.34 MiniLatTeamObjective 10 30 3 855.40 817.62 4.09 88/100 871.86 -2.35 45/100 848.30 0.79 88/100 3142.19 10 30 4 910.92 864.36 3.13 63/100 950.76 -4.82 28/100 902.24 -1.21 56/100 3360.11 10 30 5 975.78 920.09 4.27 60/100 1011.50 -4.18 32/100 959.78 0.21 53/100 3918.82 10 30 6 1031.48 976.91 3.26 60/100 1072.29 -4.58 37/100 1027.82 -1.83 51/100 4610.66 10 30 7 1088.15 1031.32 3.86 64/100 1137.31 -5.20 25/100 1083.62 0.37 59/100 5407.00 Table A.69: Complex-Task Auctions: Comparison of Team Costs for Different Numbers of Complex Targets (Overlapping Coalitions) 342 Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 10 30 3 0.00 0.01 0.01 0.01 0.00 10 30 4 0.00 0.01 0.07 0.01 0.00 10 30 5 0.00 0.03 0.10 0.03 0.00 10 30 6 0.00 0.08 0.16 0.06 0.00 10 30 7 0.00 0.94 0.24 0.66 0.00 MiniMaxTeamObjective 10 30 3 0.00 0.01 0.01 0.01 0.00 10 30 4 0.00 0.02 0.01 0.01 0.00 10 30 5 0.00 0.03 0.02 0.02 0.00 10 30 6 0.00 0.07 0.05 0.03 0.00 10 30 7 0.00 0.35 0.09 0.08 0.00 MiniLatTeamObjective 10 30 3 0.00 0.01 0.02 0.01 0.00 10 30 4 0.00 0.01 0.01 0.01 0.00 10 30 5 0.00 0.03 0.02 0.02 0.00 10 30 6 0.00 0.25 0.29 0.11 0.00 10 30 7 0.00 4.37 0.52 2.94 0.00 TableA.70: Complex-TaskAuctions: ComparisonofResponseTimesforDifferentNumbersofComplexTargets(Overlapping Coalitions) ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 419.84 344.52 16.65 91/100 350.97 14.85 87/100 344.25 16.72 93/100 1635.56 4 12 3 462.12 418.49 8.93 76/100 417.74 8.70 73/100 419.28 8.56 76/100 3652.85 6 18 3 499.51 456.17 8.07 81/100 473.89 4.33 63/100 459.39 7.26 78/100 5883.31 8 24 3 530.29 499.34 5.26 75/100 506.14 3.56 60/100 500.86 5.01 73/100 7981.37 10 30 3 568.85 540.27 4.94 74/100 545.98 3.67 58/100 547.98 3.46 74/100 9889.90 MiniMaxTeamObjective 2 6 3 198.83 186.27 5.29 76/100 191.19 2.26 60/100 187.23 4.70 75/100 917.31 4 12 3 150.63 146.02 -0.09 57/100 175.41 -21.78 30/100 147.30 -0.73 54/100 1162.61 6 18 3 140.15 132.29 1.46 58/100 150.98 -12.62 40/100 141.86 -3.30 56/100 1340.92 8 24 3 127.19 120.04 3.36 64/100 144.52 -17.59 34/100 132.56 -6.73 49/100 1447.78 10 30 3 123.78 119.15 0.09 56/100 134.83 -14.35 39/100 124.66 -4.09 53/100 1465.01 MiniLatTeamObjective 2 6 3 820.62 757.94 6.62 80/100 775.88 4.16 69/100 761.80 6.10 76/100 4911.17 4 12 3 817.97 782.64 4.12 75/100 841.10 -3.16 44/100 797.97 2.16 62/100 9289.97 6 18 3 868.90 834.11 3.66 70/100 914.64 -5.75 43/100 861.68 1.09 52/100 14114.89 8 24 3 933.81 891.00 4.21 74/100 968.32 -4.28 32/100 930.77 0.50 51/100 18269.26 10 30 3 1012.12 985.53 2.04 69/100 1062.42 -5.55 36/100 1009.17 0.84 47/100 22392.65 TableA.71: Complex-TaskAuctions: ComparisonofTeamCostsinUnknownTerrain(OverlappingCoalitions) 343 ComparisonofSKAuctions ComparisonofSKAuctions ComparisonofSKAuctions Robots Simple Complex SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) SARF andSARF(Mixed) Random Targets Targets (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 100.19 91.89 7.04 73/100 93.87 4.76 70/100 91.61 7.31 75/100 154.10 4 12 3 98.42 97.26 0.24 53/100 96.12 1.01 58/100 95.84 1.76 62/100 223.14 6 18 3 90.06 88.07 1.41 59/100 93.44 -4.98 41/100 91.51 -2.67 51/100 249.32 8 24 3 85.21 88.39 -4.90 39/100 88.03 -5.35 46/100 88.30 -4.94 37/100 257.28 10 30 3 81.46 83.60 -3.51 42/100 86.48 -7.28 33/100 83.86 -4.29 48/100 263.94 MiniMaxTeamObjective 2 6 3 95.25 92.16 2.06 59/100 92.80 0.75 56/100 91.83 2.52 62/100 154.10 4 12 3 94.95 100.78 -7.11 41/100 105.68 -13.04 31/100 97.19 -3.33 46/100 223.14 6 18 3 93.08 97.12 -6.16 39/100 102.28 -12.42 28/100 94.66 -3.14 45/100 249.32 8 24 3 87.95 91.98 -5.70 39/100 95.08 -10.19 35/100 90.39 -4.28 47/100 257.28 10 30 3 85.25 90.90 -8.46 39/100 92.42 -10.67 36/100 86.22 -2.24 44/100 263.94 MiniLatTeamObjective 2 6 3 100.19 93.79 5.44 76/100 94.30 4.70 70/100 93.92 5.19 73/100 154.10 4 12 3 98.42 94.90 2.93 65/100 96.10 1.17 54/100 94.97 2.77 63/100 223.14 6 18 3 90.06 88.91 0.34 58/100 93.21 -4.35 46/100 92.46 -3.87 48/100 249.32 8 24 3 85.21 82.77 1.74 60/100 89.85 -6.56 32/100 87.63 -3.97 45/100 257.28 10 30 3 81.46 81.03 -1.05 52/100 86.53 -7.75 34/100 85.00 -5.66 47/100 263.94 Table A.72: Complex-Task Auctions: Comparison of Numbers of Repeated Executions in Unknown Terrain (Overlapping Coalitions) Robots SimpleTargets ComplexTargets SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random MiniSumTeamObjective 2 6 3 0.04 0.73 1.05 0.75 0.10 4 12 3 0.09 0.67 0.73 0.67 0.25 6 18 3 0.15 0.66 0.68 0.68 0.41 8 24 3 0.19 0.69 0.70 0.70 0.55 10 30 3 0.24 0.70 0.72 0.71 0.70 MiniMaxTeamObjective 2 6 3 0.04 0.72 1.10 0.75 0.11 4 12 3 0.09 0.68 0.76 0.68 0.26 6 18 3 0.15 0.67 0.71 0.67 0.43 8 24 3 0.19 0.70 0.73 0.71 0.62 10 30 3 0.22 0.72 0.75 0.72 0.78 MiniLatTeamObjective 2 6 3 0.05 0.74 1.27 0.75 0.10 4 12 3 0.10 0.72 1.10 0.74 0.25 6 18 3 0.16 0.71 1.03 0.73 0.40 8 24 3 0.18 0.74 0.96 0.76 0.56 10 30 3 0.23 0.77 0.93 0.78 0.75 Table A.73: Complex-Task Auctions: Comparison of Cumulative Response Times in Unknown Terrain (Overlapping Coali- tions) 344 ComparisonofSKAuctions ComparisonofSKAuctions Robots SimpleTargets ComplexTargets SKAuctions SARF andSARF(Simple-First) SARF andSARF(Complex-First) (Simple-First) Difference(%) Dominance (Complex-First) Difference(%) Dominance 10 30 1 91.27 86.43 4.16 81/100 92.08 -3.74 60/100 10 30 2 96.47 86.62 8.06 87/100 97.95 -4.69 55/100 10 30 3 101.81 89.35 9.81 83/100 102.87 -4.31 48/100 10 30 4 105.25 94.72 7.62 73/100 107.68 -6.05 52/100 Table A.74: Comparison of Team Costs of SARF (Simple-First and Complex-First) AlgorithmswithSKAuctionsforMiniMaxTeamObjective Table A.42 tabulates the team costs of our task-allocation algorithms for disjoint coalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • The team costs of the randomized algorithm are larger than those of the other task-allocation algorithms. This observation meets our expectation since the ran- domizedalgorithmdoesnottakeanysynergyamongtargetsintoaccountwhenit allocatestargetstorobots. • TheteamcostsoftheSARF(Simple-First,Complex-First,andMixed)algorithms aresmallerthanthoseofSKAuctionsfortheMiniSumteamobjective. TheSARF (Simple-First) algorithm reduces the team costs of SK Auctions on average by 14.89percent,theSARF(Complex-First)algorithmreducestheteamcostsofSK Auctions on average by 15.17 percent, and the SARF (Mixed) algorithm reduces the team costs of SK Auctions on average by 14.83 percent. One-sided binomial tests [CB90] with normal approximation (n = 100; p = 0.5; σ 2 = 50) are per- formed to support the hypothesis that the team costs of the SARF (Mixed) algo- rithmarenotlargerthanthoseofSKAuctionswithconfidence0.99995(whichis significant). This observation meets our expectation since robots can update the visittimesoftheirallocatedtargetswhenanewtargetisallocatedtothem. • TheteamcostsoftheSARF(Mixed)algorithmaresmallerthanthoseofSKAuc- tions for the MiniMax team objective. The SARF (Mixed) algorithm reduces the teamcostsofSKAuctionsonaverageby5.93percent. Statisticaltestssupportthe 345 hypothesis that the team costs of the SARF (Mixed) algorithm are no larger than those of SK Auctions with confidence 0.9995 (which is significant). However, the team costs of the SARF (Simple-First and Complex-First) algorithms are not consistently smaller than those of SK Auctions for the MiniMax team objective. This observation does not meet our expectation but can be explained as follows: Consider the last-round allocation of the SARF algorithm for the MiniMax team objective. The resulting team cost is equal to the largest robot cost of the robots involved in the last-round allocation since the team cost of the SARF algorithm increasestheleastineachroundandtheteamcostisequaltothelargestrobotcost ofallrobots. Whentherobottarget-capacityconstraintistight,thelast-roundallo- cation of the SARF algorithm usually increases the team cost significantly since: 1) there is only d(x) eligible robots left for visiting the last unallocated (either simple or complex) target x so that the auctioneer is forced to allocate target x to these robots; and 2) Allocating targetx to these robots is usually not preferred (otherwise it would have been allocated in previous rounds). To improve our understandingofthisexplanation,weperformadditionalexperimentstocompare the team costs of the SARF (Simple-First and Complex-First) algorithms with those of SK Auctions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. We set the number of robots to 10, the numberofsimpletargetsto30,andtherobotsimple-targetcapacityto3. Wevary the number of complex targets from 1, 2, and 3 to 4 so that the robot complex- target capacity constraint is not tight. The results are reported in Table A.74. We makethefollowingobservations: – TheteamcostsoftheSARF(Simple-First)algorithmaresmallerthanthose ofSKAuctions. Thisobservationmeetsourexpectationsincethelast-round 346 r x 2 x 3 x 1 FigureA.5: Example11ofMulti-RobotRouting allocation of the SARF (Simple-First) algorithm is a complex-target allo- cation and the robot complex-target capacity constraint is not tight in our additionalexperiments. – The team costs of the SARF (Complex-First) algorithm continue to be smaller than those of SK Auctions. This observation meets our expecta- tion since the last-round allocation of the SARF (Complex-First) algorithm is a simple-target allocation and the robot simple-target capacity constraint istightinouradditionalexperiments. • The team costs of the SARF (Mixed) algorithm are smaller than those of SK Auctions for the MiniLat team objective. Statistical tests support the hypothesis that the team costs of the SARF (Mixed) algorithm are not larger than those of SK Auctions with confidence 0.9995 (which is significant). However, the team costs of the SK Auctions are very close to those of the SARF (Mixed) algorithm. The SARF (Mixed) algorithm reduces the team costs of SK Auctions on aver- age only by 0.52 percent. This observation does not meet our expectation but can be explained as follows: The difference between SK Auctions and the SARF (Mixed)algorithm is that the SARF (Mixed) algorithm can update the visit times of the allocated targets when allocating new targets to robots while robots in SK 347 BidConstruction WinnerDetermination SKAuctions SARF(Mixed) SKAuctions SARF(Mixed) Round1 Robotr: 1ontargetx 1 Robotr: 1ontargetx 1 Theauctioneerallocates Theauctioneerallocates 2ontargetx 2 2ontargetx 2 targetx 1 torobotr targetx 1 torobotr 3ontargetx 3 3ontargetx 3 Round2 Robotr: 3ontargetx 2 Robotr: 3ontargetx 2 Theauctioneerallocates Theauctioneerallocates 2ontargetx 3 2ontargetx 3 targetx 3 torobotr targetx 3 torobotr (MiniSumTeamObjective) (MiniSumTeamObjective) 4ontargetx 2 4ontargetx 2 3ontargetx 3 3ontargetx 3 (MiniMaxTeamObjective) (MiniMaxTeamObjective) 4ontargetx 2 4ontargetx 2 3ontargetx 3 3ontargetx 3 (MiniLatTeamObjective) (MiniLatTeamObjective) Round3 Robotr: 5ontargetx 2 Robotr: 4ontargetx 2 Theauctioneerallocates Theauctioneerallocates (MiniSumTeamObjective) (MiniSumTeamObjective) targetx 2 torobotr targetx 2 torobotr 8ontargetx 2 7ontargetx 2 (MiniMaxTeamObjective) (MiniMaxTeamObjective) 8ontargetx 2 8ontargetx 2 (MiniLatTeamObjective) (MiniMaxTeamObjective) TableA.75: ComparisonofSKAuctionsandSARF(Mixed)AlgorithmforExamplein FigureA.5 Auctions visit their targets in the same order in which they are allocated to them. Since the MiniLat team objective requires robots to visit targets so that the sum of the visit times of all targets is smallest, it is often effective for robots to first visit their nearest targets as they do in SK Auctions. Thus, the SARF (Mixed) algorithm usually does not need to update the visit times of the allocated targets when allocating new targets to robots so that robots in the SARF (Mixed) algo- rithm also visit the targets in the same order in which they are allocated to them. To improve our understanding of this explanation, consider an example of multi- robotroutingwithoutcomplextargetsshowninFigureA.5. Thereexistsonlyone capability. Robot r provides this capability and targets x 1 , x 2 and x 3 require it. The robot simple-target capacity is 3 so that robotr can visit all three targets for this example. Table A.75 tabulates the allocation processes of SK Auctions and the SARF (Mixed) algorithms for all three team objectives. We can see that the 348 ComparisonofSKAuctions Robots SimpleTargets ComplexTargets SKAuctions SARF andSARF(Simple-First) (Complex-First) (Complex-First) Difference(%) Dominance 2 6 1 571.47 360.52 34.46 100/100 4 12 2 980.80 582.03 39.24 100/100 6 18 3 1273.60 741.71 40.89 100/100 8 24 4 1564.88 890.93 41.96 100/100 10 30 5 1773.39 1028.97 40.89 100/100 Table A.76: Comparison of Team Costs of SARF (Complex-First) Algorithm with SK Auctions(Complex-First)forMiniLatTeamObjective SARF(Mixed)algorithmdoesnotupdatethevisittimesoftheallocatedtargetsx 1 andx 3 inthe lastround forthe MiniLatteam objectiveeventhough itcan reduce thevisittimeofthenewtargetx 2 to7ifitdoesso. • The team costs of the SARF (Complex-First) algorithm are larger than those of SK Auctions and SARF (Simple-First and Mixed) algorithms for the MiniLat team objective. This observation shows us that it is not effective to first allocate complex targets to robots for the MiniLat team objective. In order to understand theresultsbetter,weintroducethefollowingcomplex-taskallocationalgorithm: – SKAuctions(Complex-First): Thisalgorithmimplementsthegreedyauc- tion algorithm as described in Section 5.2 with the restriction that all com- plextargetsareallocatedtorobotsbeforeanysimpletarget. and perform additional experiments to compare the team costs of the SARF (Complex-First) Algorithm with those of SK Auctions (Complex-First) by using the same experimental settings of Table A.42. The results are reported in TableA.76. Wemakethefollowingobservations: – The team costs of SK Auctions (Complex-First) are much larger than those ofSKAuctions(withoutanyrestriction)showninTableA.42. Thisobserva- tionmeetsourexpectationsinceitiseffectiveforrobotstovisittheirnearest targets for the MiniLat team objective while SK Auction (Complex-First) forcesrobotstofirstmovetothelocationsofcomplextargets. 349 – The team costs of the SARF (Complex-First) algorithm are smaller than those of SA Auctions (Complex-First). The SARF (Complex-First) algo- rithm reduces the team costs of SA Auctions (Complex-First) on average by 39.49 percent. This observation meets our expectation since the SARF (Complex-First)algorithmcanchangethevisittimesoftheallocatedtargets eventhoughitfirstallocatescomplextargetstorobots. • The team costs of the SARF (Simple-First) algorithm are close to those of the SARF (Mixed) algorithm for the MiniSum and MiniLat team objectives. This observationmeetsourexpectationsincesimpletargetstendtobeallocatedbefore complex targets in the SARF (Mixed) algorithm for the MiniSum and MiniLat team objectives. When a complex target is allocated to two robots, the increase inteamcost(thatis, thesumoftheincreasesinrobotcostsofbothrobotsforthe MiniSumandMiniLatteamobjectives)isusuallylargerthantheincreaseinteam costwhenasimpletargetisallocatedtoonerobot(thatis,theincreaseintherobot costofthatsinglerobot). Table A.43 tabulates the response times of our task-allocation algorithms for dis- jointcoalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • Theresponsetimesofourtask-allocationalgorithmsareallsmall(nomatterwhat the team objective is). This observation demonstrates that the SARF (Mixed) algorithmallocatesbothcomplexandsimpletargetstorobotsefficiently. • The response times of the SARF (Complex-First) algorithm are larger than those of the SARF (Simple-First) algorithm. This observation meets our expectation: Robots that use the SARF (Simple-First) algorithm construct a reaction function foreachcomplextargetexactlyoncesincecomplextargetsareallocatedaftersim- pleonesandeachrobotcanvisitatmostonecomplextarget. However,robotsthat 350 use the SARF (Complex-First) algorithm have to update their reaction functions wheneveradditionalsimpletargetsareallocatedtothem. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms for the MiniLat team objective are larger than those for the MiniSum and MiniMaxteamobjectives. Thisobservationmeetsourexpectationsinceitismore complicated for robot r to calculate the value of the meta-functionF r,p x (t) at a givenvisittimetofcomplextargetxfortheMiniLatteamobjective,asshownon page142. Thus,ittakesmoretimetodeterminethevalueofthereactionfunction F r x (t) = min p∈P rF r,p x (t)fortheMiniLatteamobjective. Tables A.44 and A.46 tabulate the team costs of our task-allocation algorithms for disjoint coalitions inempty andindoor-like terrain, respectively. We make the follow- ingobservations: • The team costs of our task-allocation algorithms in empty terrain are all smaller than those in outdoor-like terrain shown in Table A.42, and the team costs of our task-allocation algorithms in indoor-like terrain are all larger than those in outdoor-like terrain shown in Table A.42 (no matter what the team objective is). This observation meets our expectation since robots in empty terrain do not have to circumnavigate any blocked cells and robots in indoor-like terrain have to go throughdoorsatfixedpositionstovisittargets,whereastheyonlyhavetocircum- navigategroupsofrandomlyblockedcellsinoutdoor-liketerrain. • TheteamcostsoftheSARF(Mixed)algorithmareconsistentlysmallerthanthose of SK Auctions in both empty and indoor-like terrain (no matter what the team objectiveis). TheseresultsdemonstratethattheSARF(Mixed)algorithmreduces theteamcostsofSKAuctionseffectivelyindifferentkindsofterrain. 351 TablesA.45andA.47tabulatetheresponsetimesofourtask-allocationalgorithms for disjoint coalitions in empty and indoor-like terrain, respectively. The results are almostidenticaltothoseinoutdoor-liketerrain,asshowninTableA.43andthusdemon- strate that the SARF (Mixed) algorithm allocates both complex and simple targets to robotsefficientlyindifferentkindsofterrain. Table A.48 tabulates the team costs of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain with different robot simple-target capacities. We makethefollowingobservations: • The team costs of our task-allocation algorithms all increase when the robot simple-target capacity increases (no matter what the team objective is). This observation meets our expectation since robots have to visit more simple targets whenthenumberofsimpletargetsisincreased. • TheteamcostsoftheSARF(Mixed)algorithmareconsistentlysmallerthanthose of SK Auctions for different robot simple-target capacities (no matter what the team objective is). These results demonstrate that the SARF (Mixed) algorithm reducestheteamcostsofSKAuctionseffectivelyfordifferentrobotsimple-target capacities. Table A.49 tabulates the response times of our task-allocation algorithms for dis- joint coalitions in outdoor-like terrain with different robot simple-target capacities. Wemakethefollowingobservations: • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms increase when the robot simple-target capacity increases (no matter what theteamobjectiveis). Thisobservationmeetsourexpectationsinceittakesmore times for robots to calculate their reaction functions and for the auctioneer to 352 determine the optimal visit times of complex targets when the number of targets increase. • The response times of the SARF (Complex-First) algorithm increase faster than those of the SARF (Simple-First and Mixed) algorithms (no matter what the team objective is). This observation meets our expectation since robots using the SARF (Complex-First) algorithm need to calculate more reaction functions than theotherSARFalgorithms,asexplainedbefore. • Theresponsetimesofourtask-allocationalgorithmsremainsmallwhentherobot simple-targetcapacityincreases(nomatterwhattheteamobjectiveis). Forexam- ple, the response times of the SARF (Mixed) algorithm are all smaller than one second. These results demonstrate that the SARF (Mixed) algorithm allocates complexandsimpletargetstorobotsefficientlywithdifferentrobotsimple-target capacities. Table A.50 tabulates the team costs of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain with ideal reaction-function calculations. We make thefollowingobservations: • TheteamcostsoftheSARF(Simple-First,Complex-First,andMixed)algorithms with ideal reaction-function calculations are all slightly smaller than those with approximated reaction-function calculation shown in Table A.48 (no matter what theteamobjectiveis). Thisobservationmeetsourexpectationsinceidealreaction functions typically contain more meta-functions than approximated ones. Thus, the auctioneer can find a better allocation (in terms of both coalitions and visit times) and, for a given allocation, a robot can find a better route by using ideal reactionfunctions. 353 • TheteamcostsoftheSARF(Simple-First,Complex-First,andMixed)algorithms with ideal reaction-function calculations are very close to those with approxi- mated reaction-function calculations (no matter what the team objective is). This observation demonstrates that the approximated reaction functions calculated by the Or-opt heuristic, described in Subsection 5.3.2, are very similar to the ideal ones. Table A.51 tabulates the response times of our task-allocation algorithms for dis- joint coalitions in outdoor-like terrain with ideal reaction-function calculations. We makethefollowingobservations: • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithmswithidealreaction-functioncalculationsareslightlylargerthanthosewith approximated reaction-function calculations. This observation meets our expec- tation: i)ittakesrobotsmoretimetoconstructidealreactionfunctionssincethey have to evaluate all possible routes of visiting their allocated targets; and ii) it takes the auctioneer more time to determine the optimal visit times of complex targets since the ideal reaction functions can contain more meta-functions than theapproximatedones. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms with ideal reaction-function calculations are close to those with approx- imated reaction function calculations. This observation can be explained by the approximatedreactionfunctionscalculatedbytheOr-optheuristicbeingverysim- ilartotheidealones. Table A.52 tabulates the team costs of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain with loose robot simple-target capacity constraint. Wemakethefollowingobservations: 354 Robots Simple Complex Simple-Target SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random Targets Targets Capacity Max (Min) Max (Min) Max (Min) Max (Min) Max (Min) MiniSumTeamObjective 10 30 5 ∞ 8.93 (1.16) 9.35 (0.17) 8.13 (0.17) 9.33 (0.17) 5.00 (0.00) MiniMaxTeamObjective 10 30 5 ∞ 5.88 (1.92) 5.53 (1.08) 5.91 (1.32) 5.85 (1.94) 5.00 (0.00) MiniLatTeamObjective 10 30 5 ∞ 5.88 (1.92) 5.44 (1.08) 5.54 (1.98) 5.92 (1.91) 5.00 (0.00) Table A.77: Number of Simple Targets Allocated to Robots with Robot Simple-Target Capacities∞(DisjointCoalitions) • TheteamcostsoftheSARF(Mixed)algorithmareconsistentlysmallerthanthose of SK Auctions with loose robot simple-target capacity constraints (no matter what the team objective is). These results demonstrate that the SARF (Mixed) algorithmreducestheteamcostsofSKAuctionseffectivelyevenwhentherobot simple-targetcapacityconstraintsareloose. • The team costs of the SARF (Complex-First) algorithm are smaller than those of the SARF (Simple-First) algorithm for the MiniMax team objective when the robot simple-target capacity constraints are loose. This observation meets our expectation since the last-round allocation of the SARF (Complex-First) algo- rithmisasimple-targetallocationandtherobotsimple-targetcapacityconstraints inourexperimentsarenolongertight. • The team costs of the SARF (Complex-First) algorithm are still larger than those of the SARF (Simple-First and Mixed) algorithms for the MiniLat team objec- tivewhentherobotsimple-targetcapacityconstraintsareloose. Thisobservation shows us that it is not effective to first allocate complex targets to robots for the MiniLatteamobjectiveevenwhentherobotsimple-targetcapacityconstraintsare loose. • The difference in the team costs of our task-allocation algorithms with different robotsimple-targetcapacitiesissmallwhentherobotsimple-targetcapacitiesare larger than 5 for the MiniMax and MiniLat team objectives. This observation 355 meetsourexpectation: TableA.77tabulatesthenumberofsimpletargetsallocated byourtask-allocationalgorithmstorobotswithrobotsimple-targetcapacities∞. It shows that the number of simple targets allocated to robots is on average less than6fortheMiniMaxandMiniLatteamobjectives. Thus,theallocationsofour task-allocation algorithms remain almost the same when the robot simple-target capacitiesarelargerthan5fortheMiniMaxandMiniLatteamobjectives. Table A.53 tabulates the response times of our task-allocation algorithms for dis- joint coalitions in outdoor-like terrain with loose robot simple-target capacity con- straints. Wemakethefollowingobservations: • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms increase when the robot simple-target capacity increases (no matter what the team objective is). This observation meets our expectation since there are moreeligiblerobotsforsimpletargetsandthusmorepossibleallocationsofsim- ple targets to robots that the SARF algorithms need to evaluate per round when therobotsimple-targetcapacityconstraintsareloose. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithmsremainsmallforlooserobotsimple-targetcapacityconstraints. Forexam- ple,theresponsetimesoftheSARF(Mixed)algorithmaresmallerthanthreesec- onds even when the robot simple-target capacities are infinity. This observation demonstrates us that SARF (Mixed) algorithm can allocate complex and simple targetstorobotsefficientlyevenwhentherobotsimple-targetcapacityconstraints areloose. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithmsincreasefasterwhentherobotsimple-targetcapacityincreasesfortheMin- iSum team objective than for the MiniMax and MiniLat team objectives. This 356 observationmeetsourexpectation: TableA.77showsthatthemaximalnumberof simpletargetsallocatedtorobotsismuchlargerfortheMiniSumteamobjective. Thenumberofvisitordersoftargetsthatrobotsneedtoevaluateforconstructing their reaction functions increases when there are more simple targets allocated to them. Thus, robots need more time in constructing their reaction functions for theMiniSumteamobjectivewhentherobotsimple-targetcapacityconstraintsare loose. Table A.54 tabulates the team costs of our task-allocation algorithms for disjoint coalitionsinunknownoutdoor-liketerrain. Wemakethefollowingobservations: • Theteamcostsofourtask-allocationalgorithmsinunknownterrainarealllarger thanthoseinknownterrainshowninTableA.42(nomatterwhattheteamobjec- tive is). This observation meets our expectation since robots need to repeatedly update the blockage status of cells and revise their allocations and routes when visitingtargetsinunknownterrain. • The team costs of the randomized algorithm in unknown terrain are larger than thoseoftheotheralgorithms(nomatterwhattheteamobjectiveis). Thisobserva- tionmeetsourexpectationsincetheallocationsoftargetstorobotscanbechanged completely when the randomized algorithm is executed repeatedly. A robot can be allocated targets that are far away from its current location even though it is veryclosetothepreviouslyallocatedtargets. • The team costs of the SARF (Mixed) algorithm in unknown terrain are smaller than those of SK Auctions for the MiniSum and MiniLat team objective. This observationmeetsourexpectationsincetheteamcostsoftheSARF(Mixed)algo- rithm are smaller than those of SK Auctions in known terrain. Statistical tests support the hypothesis that the team costs of the SARF (Mixed) algorithm are 357 not larger than those of SK Auctions in unknown terrain with confidence 0.9995 (whichissignificant). • The team costs of the SARF (Mixed) algorithm in unknown terrain are not con- sistentlysmallerthanthoseofSKAuctionsfortheMiniMaxteamobjective. This observationissurprisingsincethecorrespondingteamcostsoftheSARF(Mixed) algorithmaresmallerthanthoseofSKAuctionsinknownterrain. Itisfuturework tounderstandthisobservation. Table A.55 tabulates the numbers of repeated executions of our task-allocation algorithms for disjoint coalitions in unknown outdoor-like terrain. We make the fol- lowingobservations: • The numbers of repeated executions of the randomized algorithm in unknown terrain are larger than those of the other task-allocation algorithms (no matter what the team objective is). This observation meets our expectation since the team costs of the randomized algorithm are much larger than those of the other task-allocationalgorithmsinunknownterrain. • The repeated executions of the SARF (Mixed) algorithm in unknown terrain are notconsistentlysmallerthanthoseofSKAuctions(nomatterwhatteamobjective is). ThisobservationissurprisingsincetheteamcostsoftheSARF(Mixed)algo- rithm are smaller than those of SK Auctions for the MiniSum and MiniLat team objectivesinunknownterrain. Itisfutureworktounderstandthisobservation. Table A.56 tabulates the cumulative response times of our task-allocation algo- rithms for disjoint coalitions in unknown outdoor-like terrain. We make the following observations: 358 • The cumulative response times of the randomized algorithm in unknown terrain are larger than those of SK Auctions (no matter what the team objective is). This meets our expectation since the numbers of the repeated executions of the ran- domizedalgorithmarelargerthanthoseofSKAuctionsinunknownterrain. • The cumulative response times of the SARF (Simple-First, Complex-First, and Mixed) algorithms in unknown terrain are larger than those of SK Auctions (no matter what the team objective is). This observation meets our expectation since the response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithmsinknownterrainarealsolargerthanthoseofSKAuctions. TableA.57tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • The team costs of the randomized algorithm are much larger than those of the other task-allocation algorithms. This observation meets our expectation since the randomized algorithm does not take any synergy among targets into account whenitallocatestargetstorobots. • TheteamcostsoftheSARF(Simple-First,Complex-First,andMixed)algorithms aresmallerthanthoseofSKAuctionsfortheMiniSumteamobjective. TheSARF (Simple-First) algorithm reduces the team costs of SK Auctions on average by 11.16percent,theSARF(Complex-First)algorithmreducestheteamcostsofSK Auctions on average by 9.98 percent, and the SARF (Mixed) algorithm reduces the team costs of SK Auctions on average by 10.11 percent. Statistical tests sup- port the hypothesis that the team costs of the SARF (Mixed) algorithm are not larger than those of SK Auctions with confidence 0.9995 (which is significant). This observation is consistent with the results reported in Table A.42 for disjoint 359 coalitions and meets our expectation since robots can update the visit times of theirallocatedtargetswhenatargetisallocatedtothem. • The team costs of the SARF (Simple-First and Mixed) algorithms are smaller thanthoseofSKAuctionsfortheMiniMaxandMiniLatteamobjectives. Forthe MiniMax team objective, the SARF (Simple-First) algorithm reduces the team costs of SK Auctions on average by 10.20 percent, and the SARF (Mixed) algo- rithm reduces the team costs of SK Auctions on average by 4.88 percent. For the MiniLat team objective, the SARF (Simple-First) algorithm reduces the team costs of SK Auctions on average by 3.36 percent, and the SARF (Mixed) algo- rithm reduces the team costs of SK Auctions on average by 1.10 percent. Sta- tistical tests support the hypothesis that the team costs of the SARF (Simple- First,Mixed)algorithmsarenotlargerthanthoseofSKAuctionswithconfidence 0.9995(whichissignificant)fortheMiniMaxandMiniLatteamobjectives. How- ever, the team costs of the SARF (Complex-First) algorithm are not consistently smaller than those of SK Auctions for the MiniMax and MiniLat team objec- tives. This observation is not surprising for the following reasons: For the Min- iMax team objective, the robot simple-target capacity constraint is tight and the last-round allocation of the SARF (Complex-First) algorithm is a simple-target allocation; And for the MiniLat team objective, it is not effective to first allocate complextargetstorobots. Table A.58 tabulates the response times of our task-allocation algorithms for over- lappingcoalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • Theresponsetimesofourtask-allocationalgorithmsareallsmall(nomatterwhat the team objective is). This observation demonstrates that the SARF (Mixed) algorithmallocatestargetstorobotsefficientlyevenforoverlappingcoalitions. 360 • The response times of the SARF (Complex-First) algorithm are larger than those of the SARF (Simple-First) algorithm. This observation meets our expectation: Robots that use the SARF (Simple-First) algorithm construct fewer generalized reactionfunctionsthanrobotsthatusetheSARF(Complex-First)algorithmsince robots that use the SARF (Simple-First) algorithm need to construct generalized reaction functions only when complex targets are allocated to them while robots that use the SARF (Complex-First) algorithm need to construct generalized reac- tionfunctionswheneveradditionaltargetsareallocatedtothem. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms for the MiniLat team objective are larger than those for the MiniSum and MiniMaxteamobjectives. Thisobservationmeetsourexpectationsinceitismore complicated for robot r to calculate the value of the meta-functionF r,p π (t r π ) for visitingthecomplextargetsπattimest r π fortheMiniLatteamobjective,asshown onpage163. Thus,ittakesmoretimetodeterminethevalueofthereactionfunc- tionF r π (t r π ) = min p∈P r;F r,p π (t r π )fortheMiniLatteamobjective. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithmsdecreasewhentheproblemsizeincreases(nomatterwhattheteamobjec- tive is). This observation does not meet our expectation but can be explained as follows: The number of complex targets is set to 3 in our experiments. When the number of robots is small, 1) more complex targets tend to be allocated to each robot so that it takes more time for robots to construct their generalized reaction functions; and 2) more complex targets tend to be related so that it takes more timefortheauctioneertodeterminethevisittimesforcomplextargets. 361 Tables A.59 and A.61 tabulate the team costs of our task-allocation algorithms for overlapping coalitions in empty and indoor-like terrain, respectively. We make the followingobservations: • The team costs of our task-allocation algorithms in empty terrain are all smaller than those in outdoor-like terrain shown in Table A.57 , and the team costs of our task-allocation algorithms in indoor-like terrain are all larger than those in outdoor-like terrain shown in Table A.57 (no matter what the team objective is). This observation meets our expectation for the same explanation we made on the resultsshowninTablesA.44andA.46fordisjointcoalitions. • TheteamcostsoftheSARF(Mixed)algorithmareconsistentlysmallerthanthose of SK Auctions in both empty and indoor-like terrain (no matter what the team objective is). These results demonstrate that the team costs of the SARF (Mixed) algorithmaresmallerthanthoseSKAuctionsindifferentkindsofterrain. TablesA.60andA.62tabulatetheresponsetimesofourtask-allocationalgorithms foroverlappingcoalitionsinemptyandindoor-liketerrain,respectively. Theresultsare almost identical to those in outdoor-like terrain shown in Table A.58, and demonstrate thattheSARF(Mixed)algorithmallocatestargetstorobotsefficientlyindifferentkinds ofterrain. TableA.63tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitions in outdoor-like terrain with different robot simple-target capacities. We makethefollowingobservations: • The team costs of our task-allocation algorithms all increase when the robot simple-target capacity increases (no matter what the team objective is). This observationmeetsourexpectationforthesameexplanationwemadeontheresults showninTableA.48fordisjointcoalitions. 362 • The team costs of the SARF (Mixed) algorithm are all smaller than those of SK Auctions for different robot simple-target capacities (no matter what the team objective is). These results demonstrate that the team costs of the SARF (Mixed) algorithm are smaller than those of SK Auctions for different robot simple-target capacities. Table A.64 tabulates the response times of our task-allocation algorithms for over- lappingcoalitionsinoutdoor-liketerrainwithdifferentrobotsimple-targetcapacities. Wemakethefollowingobservations: • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms increase when the robot simple-target capacity increases (no matter what theteamobjectiveis). Thisobservationmeetsourexpectationsinceittakesrobots moretimetoconstructtheirgeneralizedreactionfunctionswhentheyneedtovisit moresimpletargets. • The response times of the SARF (Complex-First) algorithm increase faster than the SARF (Simple-First, Mixed) algorithms (no matter what the team objec- tive is). This observation meets our expectation since robots that use the SARF (Complex-First) algorithm need to construct more generalized reaction functions thanrobotsthatusetheotherSARFalgorithms. • Theresponsetimesofourtask-allocationalgorithmsremainsmallwhentherobot simple-target capacity increases (no matter what the team objective is). These results demonstrate that the SARF (Mixed) algorithm allocates targets to robots efficientlyfordifferentrobotsimple-targetcapacities. TableA.65tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitions in outdoor-like terrain with ideal reaction-function calculations. We make thefollowingobservations: 363 • TheteamcostsoftheSARF(Simple-First,Complex-First,andMixed)algorithms with ideal reaction-function calculations are all slightly smaller than those with approximated reaction-function calculation shown in Table A.63 (no matter what theteamobjectiveis). Thisobservationmeetsourexpectationforthesameexpla- nationwemadeontheresultsshowninTableA.50fordisjointcoalitions. • TheteamcostsoftheSARF(Simple-First,Complex-First,andMixed)algorithms with ideal reaction-function calculations are very close to those with approxi- mated reaction-function calculations (no matter what the team objective is). This observationdemonstratesthattheapproximatedreactionfunctionsconstructedby the Or-opt heuristic described in Subsection 5.4.2 are very similar to the ideal ones. TableA.66tabulatestheresponsetimesofourtask-allocationalgorithmswithover- lappingcoalitionsinoutdoor-liketerrainwithidealreaction-functioncalculation. We makethefollowingobservations: • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithmswithidealreaction-functioncalculationsareslightlylargerthanthosewith approximated reaction-function calculations shown in Table A.64. This observa- tionmeetsourexpectationforthesameexplanationwemadeontheresultsshown inTableA.51fordisjointcoalitions. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms with ideal reaction-function calculations are close to those with approxi- mated reaction-function calculations shown in Table A.64. This observation can be explained by the approximated reaction functions being very similar to the idealones. 364 Robots Simple Complex Simple-Target SKAuctions SARF(Simple-First) SARF(Complex-First) SARF(Mixed) Random Targets Targets Capacity Max (Min) Max (Min) Max (Min) Max (Min) Max (Min) MiniSumTeamObjective 10 30 5 ∞ 8.13 (0.45) 9.20 (0.36) 8.32 (0.42) 9.17 (0.36) 5.00 (0.00) MiniMaxTeamObjective 10 30 5 ∞ 5.13 (1.56) 5.01 (1.61) 5.11 (1.59) 5.07 (1.51) 5.00 (0.00) MiniLatTeamObjective 10 30 5 ∞ 5.13 (1.56) 4.87 (1.69) 5.11 (1.58) 5.00 (1.58) 5.00 (0.00) Table A.78: Number of Simple Targets Allocated to Robots with Robot Simple-Target Capacities∞(OverlappingCoalitions) TableA.67tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitions in outdoor-like terrain withlooserobotsimple-targetcapacityconstraints. Wemakethefollowingobservations: • The team costs of the SARF (Mixed) algorithm are all smaller than those of SK Auctions for loose robot simple-target capacity constraints (no matter what the team objective is). The results demonstrate that the team costs of the SARF (Mixed) algorithm are smaller than those of SK Auctions even when the robot simple-targetcapacityconstraintsareloose. • TheteamcostsoftheSARF(Complex-First)algorithmarenotsmallerthanthose of the SARF (Simple-First and Mixed) algorithms for the MiniMax and MiniLat teamobjectives. Thisobservationmeetsourexpectationforthesameexplanation wemadeontheresultsshowninTableA.52fordisjointcoalitions. • The difference in the team costs of the SARF (Simple-First, Complex-First, and Mixed) algorithms with different robot simple-target capacities becomes small when the robot simple-target capacities are larger than 5 for the MiniMax and MiniLat team objectives. This observation meets our expectation: Table A.78 tabulatesthenumberofsimpletargetsallocatedbyourtask-allocationalgorithms to robots with the robot simple-target capacities∞. It shows that the number of simple targets allocated to robots is on average less than 6 for the MiniMax and MiniLat team objectives. Thus, the allocations of our task-allocation algorithms 365 Robots SimpleTargets ComplexTargets SARF SARF SARF (Simple-First) (Complex-First) (Mixed) MiniSumTeamObjective 10 30 3 1.93 1.73 1.74 10 30 4 2.53 2.20 2.26 10 30 5 3.02 2.47 2.77 10 30 6 3.48 2.68 3.21 10 30 7 4.21 2.89 3.74 MiniMaxTeamObjective 10 30 3 2.22 1.27 1.52 10 30 4 2.89 1.70 2.06 10 30 5 3.35 2.10 2.72 10 30 6 3.92 2.53 3.17 10 30 7 4.59 3.02 3.71 MiniLatTeamObjective 10 30 3 1.62 1.27 1.43 10 30 4 2.14 1.70 1.92 10 30 5 2.72 2.11 2.44 10 30 6 3.25 2.52 2.87 10 30 7 3.84 2.98 3.38 TableA.79: NumberofRelatedComplexTargets remain almost the same when the robot simple-target capacities are larger than 5 fortheMiniMaxandMiniLatteamobjectives. TableA.68tabulatestheresponsetimesofourtask-allocationalgorithmswithover- lappingcoalitionsinoutdoor-liketerrainwithlooserobotsimple-targetcapacitycon- straints. Wemakethefollowingobservations: • The response times of SARF (Simple-First, Complex-First, and Mixed) algo- rithms remain small with loose robot simple-target capacity constraints. For example, the response times of the SARF (Mixed) algorithm are smaller than one second even when the robot simple-target capacities are∞. This observa- tion demonstrates that the SARF (Mixed) algorithm allocates targets to robots efficientlyevenwhenrobotsimple-targetcapacityconstraintsareloose. • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms increase faster with the robot simple-target capacities for the MiniSum teamobjectivethanfortheMiniMaxandMiniLatteamobjectives. Thisobserva- tionmeetsourexpectationforthesameexplanationwemadeontheresultsshown inTableA.53fordisjointcoalitions. 366 TableA.69tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitions in outdoor-like terrain with different numbers of complex targets. We make thefollowingobservations: • The team costs of the SARF (Mixed) algorithm are all smaller than those of SK Auctionsfordifferentnumbersofcomplextargets(nomatterwhattheteamobjec- tiveis). TheseresultsdemonstratethattheteamcostsoftheSARF(Mixed)algo- rithm are smaller than those of SK Auctions for different numbers of complex targets. • The team costs of the SARF (Complex-First) algorithm are larger than those of theSARF(Simple-FirstandMixed)algorithmfortheMiniMaxandMiniLatteam objectives. This observation meets our expectation for the same explanation we made on the results shown in Table A.57 when the number of complex targets is 3. Table A.70 tabulates the response times of our task-allocation algorithms for over- lapping coalitions in outdoor-like terrain with different numbers of complex targets. Wemakethefollowingobservations: • The response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms increase with the number of complex targets (no matter what the team objectiveis). Thisobservationmeetsourexpectationsince: i)robotsneedtocon- struct more generalized reaction-functions when the number of complex targets is larger; and ii) the auctioneer needs more time to determine the visit times of the related complex targets when the number of related complex targets is larger. Table A.79 tabulates the number of related complex targets for the SARF algo- rithms in our experiments. It shows that the number of related complex targets increaseswiththenumberofcomplextargets. 367 • The response times of the SARF (Simple-First) algorithm increase faster than those of the SARF (Complex-First and Mixed) algorithms (no matter what the teamobjectiveis). ThisobservationmeetsourexpectationsinceTableA.79shows thatthenumberofrelatedcomplextargetsfortheSARF(Simple-First)algorithm is much larger than those that for the SARF (Complex-First and Mixed) algo- rithms. Thus, the auctioneer needs more time for the SARF (Simple-First) algo- rithmtodeterminethevisittimesoftherelatedcomplextargets. • The response times of the SARF algorithms remain small when the number of complex targets increases (no matter what the team objective is). For example, the response times of the SARF (Mixed) algorithm are smaller than three sec- onds. This observation demonstrates that the SARF (Mixed) algorithm allocates targets to robots efficiently for different numbers of complex targets (as long as thenumberofcomplextargetsremainssmall). TableA.71tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitionsinunknownoutdoor-liketerrain. Wemakethefollowingobservations: • Theteamcostsofourtask-allocationalgorithmsinunknownterrainarealllarger thanthoseinknownterrainshowninTableA.57(nomatterwhattheteamobjec- tiveis). Thisobservationmeetsourexpectationforthesameexplanationwemade ontheresultsshowninTableA.54fordisjointcoalitions. • The team costs of the randomized algorithm in unknown terrain are larger than thoseofotheralgorithms(nomatterwhattheteamobjectiveis). Thisobservation meets our expectation for the same explanation we made on the results shown in TableA.54fordisjointcoalitions. 368 • The team costs of the SARF (Mixed) algorithm in unknown terrain are smaller than those of SK Auctions for the MiniSum and MiniLat team objectives. How- ever,theteamcostsoftheSARF(Mixed)algorithmarenotsmallerthanthoseof SKAuctionsfortheMiniMaxteamobjective. Itisfutureworktounderstandthis observation. TableA.72tabulatesthenumbersoftherepeatedexecutionsofourtask-allocation algorithms for overlapping coalitions in unknown outdoor-like terrain. We make the followingobservations: • Thenumbersoftherepeatedexecutionsoftherandomizedalgorithminunknown terrain are larger than those of the other task-allocation algorithms (no matter what the team objective is). This observation meets our expectation since the team costs of the randomized algorithm are much larger than those of the other task-allocationalgorithmsinunknownterrain,asshowninTableA.71. • The repeated executions of the SARF (Mixed) algorithm in unknown terrain can belargerthanthoseofSKAuctions(nomatterwhatteamobjectiveis). Itisfuture worktounderstandthisobservation. Table A.73 tabulates the cumulative response times of our task-allocation algo- rithms for overlapping coalitions in unknown outdoor-like terrain. We make the fol- lowingobservations: • Thecumulativeresponsetimesoftherandomizedalgorithmarelargerthanthose ofSKAuctions(nomatterwhattheteamobjectiveis). Thismeetsourexpectation since the numbers of the repeated executions of the randomized algorithm are muchlargerthanthoseofSKAuctions. 369 • The cumulative response times of the SARF (Simple-First, Complex-First, and Mixed) algorithms in unknown terrain are larger than those of SK Auctions (no matter what the team objective is). This observation meets our expectation since the response times of the SARF (Simple-First, Complex-First, and Mixed) algo- rithms in known terrain are also larger than those of SK Auctions, as shown in TableA.58. A.4 Complex-TaskNegotiationExperiments We perform the following experiments to evaluate our complex-task negotiation algo- rithms: • We evaluate the performance of our task-allocation algorithms on multi-robot routingproblemswithdisjointcoalitionsover100instancesofoutdoor-liketerrain withrandomlygeneratedlocationsofrobotsandtargets. Theinitialallocationsfor theSNRF(Complex-Only,Simple-Only,andMixed)algorithmsareproducedby theSARF(Mixed)algorithm. Wevarythenumberofrobotsfrom2,4,6,and8to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of complex targets from 1, 2, 3, and 4 to 5. We set the robot simple-target capacity to3. TheresultsarereportedinTableA.80fortheteamcosts,andTableA.81for theresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routingproblemswithdisjointcoalitionsover100instancesofemptyterrainwith randomlygeneratedlocationsofrobotsandtargets. Theinitialallocationsforthe SNRF(Complex-Only,Simple-Only,andMixed)algorithmsareproducedbythe SARF (Mixed) algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of 370 complex targets from 1, 2, 3, and 4 to 5. We set the robot simple-target capacity to3. TheresultsarereportedinTableA.82fortheteamcosts,andTableA.83for theresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routingproblemswithdisjointcoalitionsover100instancesofindoor-liketerrain with randomly generated locations of robots and targets. The initial allocations fortheSNRF(Complex-Only,Simple-Only,andMixed)algorithmsareproduced by the SARF (Mixed) algorithm. We vary the number of robots from 2, 4, 6, and 8to10,thenumberofsimpletargetsfrom6,12,18,and24to30,andthenumber ofcomplextargetsfrom1,2,3,and4to5. Wesettherobotsimple-targetcapacity to3. TheresultsarereportedinTableA.84fortheteamcosts,andTableA.85for theresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithdifferentrobot simple-target capacities on multi-robot routing problems with disjoint coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. The initial allocations for the SNRF (Complex-Only, Simple- Only, and Mixed) algorithms are produced by the SARF (Mixed) algorithm. We set the number of robots to 10 and the number of complex targets to 5. We vary thenumberofsimpletargetsfrom20,30,and40to50andtherobotsimple-target capacityfrom2,3,and4to5. TheresultsarereportedinTableA.86fortheteam costs,andTableA.87fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithidealreaction- functioncalculationonmulti-robotroutingproblemswithdisjointcoalitionsover 100instancesofoutdoor-liketerrainwithrandomlygeneratedlocationsofrobots and targets. The initial allocations for the SNRF (Complex-Only, Simple-Only, 371 and Mixed) algorithms are produced by the SARF (Mixed) algorithm. We set the number of robots to 10 and the number of complex targets to 5. We vary the number of simple targets from 20, 30, and 40 to 50 and the robot simple-target capacityfrom2,3,and4to5. TheresultsarereportedinTableA.88fortheteam costsandTableA.89fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms with loose robot simple-target capacity constraint on multi-robot routing problems with disjoint coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. The initial allocations for the SNRF (Complex- Only, Simple-Only, and Mixed) algorithms are produced by the SARF (Mixed) algorithm. Wesetthenumberofrobotsto10,thenumberofsimpletargetsto30, andthenumberofcomplextargetsto5. Wevarytherobotsimple-targetcapacity from 3, 4, 5, 6, and 7 to∞. The results are reported in Table A.90 for the team costs,andTableA.91fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routingproblemswithdisjointcoalitionsover100instancesofoutdoor-liketerrain withrandomlygeneratedlocationsofrobotsandtargets. Theinitialallocationsfor theSNRF(Complex-Only,Simple-Only,andMixed)algorithmsareproducedby the randomized algorithm. We varied the number of robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of complex targets from 1, 2, 3, and 4 to 5. We set the robot simple-target capacity to3. TheresultsarereportedinTableA.92fortheteamcosts,andTableA.93for theresponsetimes. • We evaluate the performance of our task-allocation algorithms in unknown ter- rain on multi-robot routing problems with disjoint coalitions over 100 instances 372 of outdoor-like terrain with randomly generated locations of robots and targets. The initial allocations for the SNRF (Complex-Only, Simple-Only, and Mixed) algorithmsareproducedbytheSARF(Mixed)algorithm. Wevarythenumberof robots from 2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30, and the number of complex targets from 1, 2, 3, and 4 to 5. We set the robot simple-target capacity to 3. The results are reported in Table A.94 for the team costs, Table A.95 for the numbers of repeated executions, and Table A.96 forthecumulativeresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlapping coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. The initial allo- cations for the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms are producedbytheSARF(Mixed)algorithm. Wevarythenumberofrobotsfrom2, 4, 6, and 8 to 10, the number of simple targets from 6, 12, 18, and 24 to 30. We set the number of complex targets to 3, the robot simple-target capacity to 3, and the boundR of the number of related robots to 10. The results are reported in TableA.97fortheteamcosts,andTableA.98fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlappingcoalitions over100 instances of empty terrain with randomly generated locations of robots and targets. The initial allocations fortheSNRF(Complex-Only,Simple-Only,andMixed)algorithmsareproduced by the SARF (Mixed) algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10 and the number of simple targets from 6, 12, 18, and 24 to 30. We set the number of complex targets to 3, the robot simple-target capacity to 3, and 373 the boundR of the number of related robots to 10. The results are reported in TableA.99fortheteamcosts,andTableA.100fortheresponsetimes. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlapping coalitions over 100 instances of indoor-like terrain with randomly generated locations of robots and targets. The initial allo- cations for the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms are producedbytheSARF(Mixed)algorithm. Wevarythenumberofrobotsfrom2, 4, 6, and 8 to 10 and the number of simple targets from 6, 12, 18, and 24 to 30. We set the number of complex targets to 3, the robot simple-target capacity to 3, andtheboundRofthenumberofrelatedrobotsto10. Theresultsarereportedin TableA.101fortheteamcosts,andTableA.102fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithdifferentrobot simple-target capacities on multi-robot routing problems with overlapping coali- tions over 100 instances of outdoor-like terrain with randomly generated loca- tions of robots and targets. The initial allocations for the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms are produced by the SARF (Mixed) algo- rithm. We vary the number of simple targets from 20, 30, and 40 to 50 and the robot simple-target capacity from 2, 3, and 4 to 5. We set the number of robots to 10, the number of complex targets to 3, and the boundR of the number of related robots to 10. The results are reported in Table A.103 for the team costs, andTableA.104fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithidealreaction- function calculation on multi-robot routing problems with overlapping coalitions over 100 instances of outdoor-like terrain with randomly generated locations of 374 robots and targets. The initial allocations for the SNRF (Complex-Only, Simple- Only, and Mixed) algorithms are produced by the SARF (Mixed) algorithm. We vary the number of simple targets from 20, 30, and 40 to 50 and the number of robot simple-target capacity from 2, 3, and 4 to 5. We set the number of robots to 10, the number of complex targets to 3, and the boundR of the number of related robots to 10. The results are reported in Table A.103 for the team costs, andTableA.104fortheresponsetimes. • We evaluated the performance of our task-allocation algorithms with loose robot simple-target capacity constraint on multi-robot routing problems with overlap- pingcoalitionsover100instancesofoutdoor-liketerrainwithrandomlygenerated locations of robots and targets. The initial allocations for the SNRF (Complex- Only, Simple-Only, and Mixed) algorithms are produced by the SARF (Mixed) algorithm. We vary the robot simple-target capacity from 3, 4, 5, 6, and 7 to∞. Wesetthenumberofrobotsto10,thenumberofsimpletargetsto30,thenumber of complex targets to 3, and the boundR of the number of related robots to 10. TheresultsarereportedinTableA.107fortheteamcostsandTableA.108forthe responsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmswithdifferentnum- ber of complex targets on multi-robot routing problems with overlapping coali- tions over 100 instances of outdoor-like terrain with randomly generated loca- tions of robots and targets. The initial allocations for the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms are produced by the SARF (Mixed) algo- rithm. Wevarythenumberofcomplextargetsfrom3,4,5,and6to7. Wesetthe numberofrobotsto10,thenumberofsimpletargetsto30,therobotsimple-target capacity to 3, and the boundR of the number of related robots to 10. The results 375 are reported in Table A.109 for the team costs, and Table A.110 for the response times. • We evaluate the performance of our task-allocation algorithms on multi-robot routing problems with overlapping coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. The initial allo- cations for the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms are produced by the randomized algorithm. We vary the number of robots from 2, 4, 6, and 8 to 10 and the number of simple targets from 6, 12, 18, and 24 to 30. We set the number of complex targets to 3, the robot simple-target capacity to 3, and the boundR of the number of related robots to 10. The results are reported in TableA.111fortheteamcosts,andTableA.112fortheresponsetimes. • Weevaluatetheperformanceofourtask-allocationalgorithmsinunknownterrain on multi-robot routing problems with overlapping coalitions over 100 instances of outdoor-like terrain with randomly generated locations of robots and targets. Wevarythenumberof robotsfrom 2, 4, 6, and8to 10and thenumberof simple targetsfrom6,12,18,and24to30. Wesetthenumberofcomplextargetsto3,the robotsimple-targetcapacityto3,andtheboundRofthenumberofrelatedrobots to10. TheresultsarereportedinTableA.113fortheteamcosts,TableA.114for thenumbersofrepeatedexecutions,andTableA.115forthecumulativeresponse times. Wenowexaminetheseresultsinmoredetail: 376 ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 193.76 193.76 0.00 100/100 185.57 4.03 100/100 185.57 0.00 100/100 246.55 290.04 4 12 2 313.83 306.08 2.26 100/100 286.50 6.20 87/100 283.12 1.04 97/100 379.23 581.62 6 18 3 418.77 400.79 4.06 100/100 372.79 6.61 88/100 363.34 2.34 95/100 487.97 892.70 8 24 4 514.52 485.38 5.41 100/100 456.71 5.70 86/100 440.31 3.48 93/100 593.06 1179.98 10 30 5 577.35 549.53 4.54 100/100 514.49 6.31 88/100 502.05 2.28 91/100 660.18 1471.14 MiniMaxTeamObjective 2 6 1 109.06 109.06 0.00 100/100 98.62 8.82 100/100 98.62 0.00 100/100 119.50 163.58 4 12 2 108.23 101.74 5.42 100/100 84.74 15.00 91/100 83.36 1.18 91/100 115.44 182.15 6 18 3 104.84 95.44 8.16 100/100 79.07 15.52 94/100 77.14 2.22 88/100 114.55 194.10 8 24 4 107.12 93.28 11.89 100/100 77.96 14.54 86/100 74.22 4.00 77/100 112.99 197.27 10 30 5 109.51 93.74 13.90 100/100 75.74 16.98 86/100 72.06 4.06 75/100 115.31 203.49 MiniLatTeamObjective 2 6 1 355.37 355.37 0.00 100/100 337.48 4.66 100/100 337.48 0.00 100/100 357.99 703.53 4 12 2 562.08 553.69 1.46 100/100 518.71 6.09 92/100 513.88 0.91 98/100 565.40 1413.96 6 18 3 723.77 709.80 1.90 100/100 661.08 6.49 93/100 651.10 1.47 91/100 728.57 2183.76 8 24 4 866.46 845.28 2.44 100/100 786.96 6.51 91/100 772.72 1.82 97/100 870.77 2884.06 10 30 5 996.26 972.10 2.38 100/100 901.92 7.03 97/100 884.73 1.83 92/100 1000.40 3600.49 TableA.80: Complex-TaskNegotiations: ComparisonofTeamCostsinOutdoor-LikeTerrain(DisjointCoalitions) Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.05 0.05 0.00 0.00 6 18 3 0.00 0.02 0.17 0.18 0.00 0.00 8 24 4 0.01 0.04 0.37 0.44 0.00 0.00 10 30 5 0.01 0.08 0.71 0.78 0.00 0.00 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.05 0.05 0.00 0.00 6 18 3 0.00 0.02 0.15 0.16 0.00 0.00 8 24 4 0.01 0.04 0.36 0.37 0.00 0.00 10 30 5 0.01 0.07 0.61 0.68 0.00 0.00 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.05 0.06 0.00 0.00 6 18 3 0.00 0.02 0.17 0.20 0.00 0.00 8 24 4 0.01 0.04 0.41 0.51 0.00 0.00 10 30 5 0.02 0.07 0.86 1.06 0.00 0.00 TableA.81: Complex-TaskNegotiations: ComparisonofResponseTimesinOutdoor-LikeTerrain(DisjointCoalitions) 377 ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 178.57 178.57 0.00 100/100 169.01 5.21 100/100 169.01 0.00 100/100 226.95 279.43 4 12 2 292.14 280.48 3.75 100/100 268.55 3.75 81/100 259.93 2.94 97/100 349.18 563.21 6 18 3 376.96 360.21 4.28 100/100 338.56 5.77 83/100 330.23 2.34 95/100 437.01 839.71 8 24 4 457.07 431.35 5.33 100/100 406.84 5.58 83/100 394.22 2.89 95/100 531.33 1116.10 10 30 5 523.79 497.64 4.88 100/100 463.25 6.67 85/100 448.14 3.16 95/100 592.33 1404.23 MiniMaxTeamObjective 2 6 1 99.63 99.63 0.00 100/100 89.00 9.78 100/100 89.00 0.00 100/100 107.79 157.10 4 12 2 98.99 91.49 7.10 100/100 78.85 12.43 89/100 77.04 2.06 93/100 104.89 176.42 6 18 3 96.76 85.61 10.87 100/100 73.31 12.77 86/100 70.74 3.01 88/100 102.12 186.63 8 24 4 95.91 83.98 12.16 100/100 71.34 13.33 85/100 67.67 4.74 87/100 101.83 191.44 10 30 5 98.49 85.04 12.87 100/100 69.99 15.64 83/100 65.72 5.43 84/100 103.45 195.53 MiniLatTeamObjective 2 6 1 328.44 328.44 0.00 100/100 311.36 4.79 100/100 311.36 0.00 100/100 329.84 682.61 4 12 2 503.75 493.97 1.84 100/100 466.02 5.27 85/100 461.11 0.97 96/100 504.77 1372.53 6 18 3 646.04 631.05 2.25 100/100 592.09 5.83 90/100 579.55 2.07 96/100 649.02 2050.01 8 24 4 780.10 755.45 3.06 100/100 710.17 5.81 94/100 696.08 1.92 92/100 786.35 2716.89 10 30 5 892.21 870.30 2.41 100/100 799.95 7.79 96/100 788.66 1.41 89/100 898.67 3437.42 TableA.82: Complex-TaskNegotiations: ComparisonofTeamCostsinEmptyTerrain(DisjointCoalitions) Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.04 0.05 0.00 0.00 6 18 3 0.00 0.02 0.14 0.16 0.00 0.00 8 24 4 0.01 0.04 0.35 0.39 0.00 0.00 10 30 5 0.01 0.08 0.67 0.79 0.00 0.00 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.05 0.05 0.00 0.00 6 18 3 0.00 0.02 0.15 0.17 0.00 0.00 8 24 4 0.01 0.04 0.31 0.37 0.00 0.00 10 30 5 0.01 0.07 0.57 0.67 0.00 0.00 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.05 0.05 0.00 0.00 6 18 3 0.00 0.02 0.17 0.20 0.00 0.00 8 24 4 0.01 0.04 0.40 0.49 0.00 0.00 10 30 5 0.02 0.08 0.86 0.96 0.00 0.00 TableA.83: Complex-TaskNegotiations: ComparisonofResponseTimesinEmptyTerrain(DisjointCoalitions) 378 ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 281.28 281.28 0.00 100/100 269.29 3.83 100/100 269.29 0.00 100/100 354.07 444.70 4 12 2 459.74 444.82 2.93 100/100 421.19 4.92 81/100 410.57 2.23 97/100 543.33 873.03 6 18 3 589.09 570.08 3.11 100/100 533.78 6.14 88/100 522.19 2.02 96/100 687.89 1344.70 8 24 4 709.59 681.19 3.86 100/100 638.05 5.99 81/100 619.48 2.77 94/100 815.11 1740.61 10 30 5 807.62 771.28 4.39 100/100 718.54 6.61 88/100 693.79 3.39 93/100 913.02 2203.08 MiniMaxTeamObjective 2 6 1 156.33 156.33 0.00 100/100 145.71 6.08 100/100 145.71 0.00 100/100 167.36 250.35 4 12 2 155.57 145.01 6.23 100/100 129.98 9.16 82/100 125.71 2.90 95/100 162.42 271.40 6 18 3 154.02 137.06 10.31 100/100 117.64 12.50 84/100 112.72 3.34 89/100 160.26 292.36 8 24 4 150.58 134.89 9.79 100/100 113.49 14.27 87/100 108.56 3.83 85/100 158.44 291.69 10 30 5 153.46 135.13 11.79 100/100 114.51 12.91 79/100 105.71 7.30 87/100 160.89 304.48 MiniLatTeamObjective 2 6 1 518.13 518.13 0.00 100/100 500.59 3.01 100/100 500.59 0.00 100/100 521.40 1100.24 4 12 2 794.25 779.66 1.74 100/100 749.68 3.51 86/100 737.76 1.58 99/100 797.16 2155.33 6 18 3 1006.67 979.35 2.58 100/100 924.10 5.30 88/100 907.54 1.75 97/100 1016.89 3292.32 8 24 4 1195.42 1173.58 1.77 100/100 1092.57 6.63 96/100 1072.25 1.82 97/100 1203.52 4240.61 10 30 5 1367.34 1325.01 3.05 100/100 1234.76 6.70 91/100 1207.02 2.12 93/100 1366.16 5402.24 TableA.84: Complex-TaskNegotiations: ComparisonofTeamCostsinIndoor-LikeTerrain(DisjointCoalitions) Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.04 0.05 0.00 0.00 6 18 3 0.00 0.02 0.15 0.16 0.00 0.00 8 24 4 0.01 0.04 0.35 0.41 0.00 0.00 10 30 5 0.01 0.07 0.65 0.79 0.00 0.00 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.04 0.05 0.00 0.00 6 18 3 0.00 0.02 0.16 0.17 0.00 0.00 8 24 4 0.01 0.04 0.32 0.35 0.00 0.00 10 30 5 0.01 0.07 0.60 0.70 0.00 0.00 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.04 0.05 0.00 0.00 6 18 3 0.00 0.02 0.17 0.21 0.00 0.00 8 24 4 0.01 0.04 0.38 0.46 0.00 0.00 10 30 5 0.02 0.07 0.82 0.99 0.00 0.00 TableA.85: Complex-TaskNegotiations: ComparisonofResponseTimesinIndoor-LikeTerrain(DisjointCoalitions) 379 Simple- ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Target SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets Capacities (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 5 2 517.68 487.15 5.69 100/100 459.47 5.52 88/100 445.28 2.96 96/100 574.30 1142.38 10 30 5 3 577.35 549.53 4.54 100/100 514.49 6.31 88/100 502.05 2.28 91/100 660.18 1471.14 10 40 5 4 629.27 598.05 4.85 100/100 565.22 5.32 85/100 541.59 4.07 99/100 725.13 1833.78 10 50 5 5 673.05 648.30 3.56 100/100 600.78 7.24 90/100 587.11 2.21 93/100 796.32 2165.77 MiniMaxTeamObjective 10 20 5 2 97.34 80.10 16.22 100/100 67.67 14.24 86/100 65.33 2.73 80/100 99.70 164.64 10 30 5 3 109.51 93.74 13.90 100/100 75.74 16.98 86/100 72.06 4.06 75/100 115.31 203.49 10 40 5 4 117.03 102.60 11.57 100/100 81.90 18.94 95/100 78.46 3.89 86/100 124.67 246.18 10 50 5 5 119.54 106.93 9.84 100/100 87.97 16.58 92/100 85.11 2.78 81/100 131.63 287.25 MiniLatTeamObjective 10 20 5 2 685.25 664.58 2.86 100/100 626.40 5.52 87/100 612.18 2.20 93/100 687.16 2162.04 10 30 5 3 996.26 972.10 2.38 100/100 901.92 7.03 97/100 884.73 1.83 92/100 1000.40 3600.49 10 40 5 4 1341.16 1308.83 2.41 100/100 1200.97 8.00 98/100 1180.15 1.68 90/100 1353.17 5480.09 10 50 5 5 1708.96 1672.99 2.05 100/100 1523.42 8.74 99/100 1503.17 1.30 87/100 1735.43 7598.29 Table A.86: Complex-Task Negotiations: Comparison of Team Costs for Different Robot Simple-Target Capacities (Disjoint Coalitions) Robots SimpleTargets ComplexTargets Simple-TargetCapacities SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 20 5 2 0.00 0.01 0.07 0.06 0.00 0.00 10 30 5 3 0.01 0.08 0.71 0.78 0.00 0.00 10 40 5 4 0.10 0.46 7.74 8.25 0.00 0.00 10 50 5 5 0.37 1.27 17.92 19.33 0.00 0.00 MiniMaxTeamObjective 10 20 5 2 0.00 0.01 0.05 0.05 0.00 0.00 10 30 5 3 0.01 0.07 0.61 0.68 0.00 0.00 10 40 5 4 0.08 0.35 6.00 6.24 0.00 0.00 10 50 5 5 0.35 1.57 18.83 20.33 0.00 0.00 MiniLatTeamObjective 10 20 5 2 0.00 0.01 0.06 0.08 0.00 0.00 10 30 5 3 0.02 0.07 0.86 1.06 0.00 0.00 10 40 5 4 0.11 0.41 9.29 10.91 0.00 0.00 10 50 5 5 0.47 1.55 33.13 36.87 0.00 0.00 Table A.87: Complex-Task Negotiations: Comparison of Response Times for Different Robot Simple-Target Capacities (Dis- jointCoalitions) 380 Simple- ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Target SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets Capacities (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 5 2 517.62 487.10 5.69 100/100 459.79 5.45 87/100 445.24 3.03 96/100 574.30 1142.38 10 30 5 3 577.11 549.52 4.50 100/100 514.32 6.33 87/100 501.80 2.30 91/100 660.18 1471.14 10 40 5 4 627.80 597.22 4.76 100/100 563.43 5.50 86/100 541.78 3.69 98/100 725.13 1833.78 10 50 5 5 668.32 641.77 3.83 100/100 598.92 6.63 92/100 583.28 2.52 95/100 796.32 2165.77 MiniMaxTeamObjective 10 20 5 2 97.33 80.10 16.21 100/100 67.76 14.13 86/100 65.25 2.97 80/100 99.70 164.64 10 30 5 3 109.51 93.68 13.96 100/100 75.72 16.95 85/100 72.02 4.06 75/100 115.31 203.49 10 40 5 4 117.36 102.69 11.73 100/100 82.06 18.79 95/100 78.91 3.59 83/100 124.67 246.18 10 50 5 5 118.88 106.82 9.63 100/100 88.92 15.38 90/100 85.15 3.85 87/100 131.63 287.25 MiniLatTeamObjective 10 20 5 2 684.80 663.89 2.90 100/100 625.97 5.48 87/100 611.64 2.23 94/100 687.16 2162.04 10 30 5 3 993.99 971.81 2.20 100/100 901.16 7.09 96/100 885.10 1.71 93/100 1000.40 3600.49 10 40 5 4 1334.43 1305.39 2.18 100/100 1198.57 7.94 99/100 1178.40 1.64 90/100 1353.17 5480.09 10 50 5 5 1703.08 1663.93 2.26 100/100 1511.40 8.93 99/100 1496.48 1.00 84/100 1735.43 7598.29 TableA.88: Complex-TaskNegotiations: ComparisonofTeamCostsforIdealReaction-FunctionCalculations(DisjointCoali- tions) Robots SimpleTargets ComplexTargets Simple-TargetCapacities SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 20 5 2 0.00 0.01 0.07 0.06 0.00 0.00 10 30 5 3 0.02 0.09 0.80 0.91 0.00 0.00 10 40 5 4 0.11 0.47 8.04 8.55 0.00 0.00 10 50 5 5 0.44 1.33 18.98 20.45 0.00 0.00 MiniMaxTeamObjective 10 20 5 2 0.00 0.01 0.05 0.05 0.00 0.00 10 30 5 3 0.02 0.07 0.62 0.70 0.00 0.00 10 40 5 4 0.10 0.37 6.15 6.40 0.00 0.00 10 50 5 5 0.41 1.61 19.83 22.51 0.00 0.00 MiniLatTeamObjective 10 20 5 2 0.00 0.01 0.06 0.08 0.00 0.00 10 30 5 3 0.03 0.07 0.90 1.17 0.00 0.00 10 40 5 4 0.13 0.45 9.59 11.43 0.00 0.00 10 50 5 5 0.57 1.67 36.87 40.13 0.00 0.00 Table A.89: Complex-Task Negotiations: Comparison of Response Times for Ideal Reaction-Function Calculations (Disjoint Coalitions) 381 Simple- ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Target SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets Capacities (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 30 5 3 577.35 549.71 4.52 100/100 514.39 6.35 87/100 501.70 2.33 92/100 660.18 1471.14 10 30 5 4 515.18 483.57 5.86 100/100 465.44 3.64 77/100 447.52 3.64 95/100 576.31 1472.93 10 30 5 5 496.52 467.24 5.58 100/100 455.45 2.47 70/100 437.50 3.77 93/100 559.93 1462.98 10 30 5 6 485.17 459.20 5.12 100/100 449.84 1.92 66/100 433.63 3.36 89/100 552.73 1462.98 10 30 5 7 482.01 452.64 5.82 100/100 447.94 0.86 61/100 431.16 3.61 94/100 538.71 1462.98 10 30 5 ∞ 475.00 447.86 5.52 100/100 448.60 -0.33 54/100 429.84 4.00 96/100 549.75 1462.98 MiniMaxTeamObjective 10 30 5 3 109.51 93.74 13.90 100/100 75.74 16.98 86/100 72.06 4.06 75/100 115.31 203.49 10 30 5 4 90.48 71.28 19.61 100/100 68.88 2.69 60/100 64.68 5.56 85/100 94.18 230.26 10 30 5 5 89.72 68.76 21.63 100/100 68.37 -0.06 60/100 63.11 6.80 89/100 94.06 248.89 10 30 5 6 89.76 67.72 22.71 100/100 68.22 -1.46 59/100 62.45 7.43 89/100 94.54 248.89 10 30 5 7 89.83 67.96 22.67 100/100 68.98 -2.18 55/100 62.71 7.80 84/100 94.77 248.89 10 30 5 ∞ 89.54 68.19 22.23 100/100 68.78 -1.64 58/100 62.71 7.57 84/100 95.16 248.89 MiniLatTeamObjective 10 30 5 3 996.26 972.10 2.38 100/100 901.92 7.03 97/100 884.73 1.83 92/100 1000.40 3600.49 10 30 5 4 865.82 855.44 1.21 100/100 819.44 3.99 95/100 809.02 1.26 97/100 871.95 3885.91 10 30 5 5 847.45 835.47 1.42 100/100 805.89 3.34 88/100 795.02 1.35 90/100 850.50 4002.94 10 30 5 6 842.11 829.34 1.52 100/100 802.35 3.03 85/100 791.37 1.37 94/100 844.04 4002.94 10 30 5 7 840.00 827.40 1.51 100/100 801.92 2.85 84/100 789.70 1.51 95/100 842.67 4002.94 10 30 5 ∞ 839.61 827.12 1.50 100/100 801.54 2.86 84/100 790.17 1.40 95/100 842.80 4002.94 Table A.90: Complex-Task Negotiations: Comparison of Team Costs for Loose Robot Simple-Target Capacity Constraints (DisjointCoalitions) Robots SimpleTargets ComplexTargets Simple-TargetCapacities SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 30 5 3 0.01 0.08 0.71 0.78 0.00 0.00 10 30 5 4 0.05 0.25 2.52 2.97 0.00 0.00 10 30 5 5 0.14 0.39 4.16 4.71 0.00 0.00 10 30 5 6 0.36 0.58 6.50 7.14 0.00 0.00 10 30 5 7 0.54 0.87 7.99 9.47 0.00 0.00 10 30 5 ∞ 2.00 2.42 10.12 11.53 0.00 0.00 MiniMaxTeamObjective 10 30 5 3 0.01 0.07 0.64 0.73 0.00 0.00 10 30 5 4 0.04 0.18 1.94 2.10 0.00 0.00 10 30 5 5 0.06 0.29 3.50 3.86 0.00 0.00 10 30 5 6 0.09 0.37 4.34 4.92 0.00 0.00 10 30 5 7 0.10 0.43 6.00 6.86 0.00 0.00 10 30 5 ∞ 0.11 0.45 6.42 7.17 0.00 0.00 MiniLatTeamObjective 10 30 5 3 0.02 0.07 0.86 1.06 0.00 0.00 10 30 5 4 0.04 0.20 2.36 2.85 0.00 0.00 10 30 5 5 0.08 0.32 4.01 4.37 0.00 0.00 10 30 5 6 0.12 0.44 5.43 6.06 0.00 0.00 10 30 5 7 0.13 0.49 7.41 7.97 0.00 0.00 10 30 5 ∞ 0.14 0.55 8.07 8.60 0.00 0.00 TableA.91: Complex-TaskNegotiations: ComparisonofResponseTimesforLooseRobotSimple-TargetCapacityConstraints (DisjointCoalitions) 382 ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Random SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions SARF Targets Targets (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance (Mixed) MiniSumTeamObjective 2 6 1 290.04 213.61 25.46 100/100 186.22 12.54 100/100 186.22 0.00 100/100 246.55 193.76 4 12 2 581.62 397.78 31.13 100/100 307.88 22.15 98/100 295.33 3.86 92/100 379.23 313.83 6 18 3 892.70 581.02 34.56 100/100 418.60 27.65 100/100 396.94 4.89 87/100 487.97 418.77 8 24 4 1179.98 762.21 35.23 100/100 531.96 29.94 100/100 492.61 7.19 91/100 593.06 514.52 10 30 5 1471.14 936.15 36.17 100/100 632.01 32.28 100/100 572.13 9.20 91/100 660.18 577.35 MiniMaxTeamObjective 2 6 1 163.58 115.21 28.53 100/100 99.59 13.13 100/100 99.59 0.00 100/100 119.50 109.06 4 12 2 182.15 114.23 36.19 100/100 90.67 20.24 97/100 87.52 2.92 88/100 115.44 108.23 6 18 3 194.10 118.23 38.53 100/100 91.17 22.49 99/100 86.74 3.90 76/100 114.55 104.84 8 24 4 197.27 117.06 40.09 100/100 92.97 20.14 99/100 89.81 2.83 72/100 112.99 107.12 10 30 5 203.49 120.46 40.21 100/100 97.03 18.91 97/100 93.50 2.57 64/100 115.31 109.51 MiniLatTeamObjective 2 6 1 703.53 415.13 39.19 100/100 343.02 16.47 100/100 343.02 0.00 100/100 357.99 355.37 4 12 2 1413.96 804.78 42.49 100/100 543.39 32.00 100/100 527.99 2.76 95/100 565.40 562.08 6 18 3 2183.76 1193.82 44.83 100/100 716.74 39.65 100/100 681.37 4.85 89/100 728.57 723.77 8 24 4 2884.06 1566.75 45.41 100/100 866.85 44.28 100/100 813.31 6.16 91/100 870.77 866.46 10 30 5 3600.49 1940.72 45.82 100/100 1005.66 47.96 100/100 940.63 6.41 86/100 1000.40 996.26 TableA.92: Complex-TaskNegotiations: ComparisonofTeamCostsforRandomizedInitialAllocations(DisjointCoalitions) Robots SimpleTargets ComplexTargets Random SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions SARF(Mixed) MiniSumTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.08 0.09 0.00 0.00 6 18 3 0.00 0.02 0.35 0.41 0.00 0.00 8 24 4 0.00 0.04 0.93 1.11 0.00 0.01 10 30 5 0.00 0.10 1.92 2.27 0.00 0.01 MiniMaxTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.08 0.08 0.00 0.00 6 18 3 0.00 0.02 0.25 0.30 0.00 0.00 8 24 4 0.00 0.03 0.57 0.61 0.00 0.01 10 30 5 0.00 0.08 0.91 1.02 0.00 0.01 MiniLatTeamObjective 2 6 1 0.00 0.00 0.00 0.00 0.00 0.00 4 12 2 0.00 0.00 0.09 0.11 0.00 0.00 6 18 3 0.00 0.02 0.43 0.52 0.00 0.00 8 24 4 0.00 0.04 1.14 1.38 0.00 0.01 10 30 5 0.00 0.10 2.47 3.16 0.00 0.02 Table A.93: Complex-Task Negotiations: Comparison of Response Times for Randomized Initial Allocations (Disjoint Coali- tions) 383 ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 255.41 255.41 0.00 100/100 241.50 4.87 86/100 241.50 0.00 100/100 304.38 1087.37 4 12 2 402.79 399.93 0.09 75/100 371.85 6.04 79/100 359.09 2.69 83/100 463.84 3065.27 6 18 3 533.97 516.74 2.91 72/100 498.75 2.86 68/100 481.99 2.88 76/100 588.29 5230.18 8 24 4 642.08 629.07 1.55 60/100 593.64 4.70 60/100 574.82 2.97 74/100 694.05 7155.15 10 30 5 726.20 703.13 2.71 68/100 677.59 3.41 66/100 658.42 2.50 67/100 767.94 9646.11 MiniMaxTeamObjective 2 6 1 142.90 142.90 0.00 100/100 129.65 8.17 86/100 129.65 0.00 100/100 151.25 650.79 4 12 2 146.87 141.87 2.57 75/100 117.70 13.40 80/100 114.16 1.97 73/100 141.02 1002.26 6 18 3 146.21 135.73 5.10 78/100 116.54 11.20 77/100 113.33 0.94 60/100 146.71 1257.41 8 24 4 141.66 128.34 6.69 66/100 115.29 7.39 63/100 109.04 3.47 65/100 137.17 1338.25 10 30 5 142.16 127.65 6.87 67/100 112.86 9.08 70/100 106.35 4.56 61/100 136.78 1455.60 MiniLatTeamObjective 2 6 1 451.36 451.36 0.00 100/100 429.41 4.32 94/100 429.41 0.00 100/100 459.14 2450.61 4 12 2 687.19 684.27 0.31 83/100 653.76 4.21 71/100 651.36 0.24 83/100 695.67 6991.06 6 18 3 885.46 869.64 1.70 73/100 844.51 2.41 63/100 828.36 1.60 83/100 906.70 12777.91 8 24 4 1059.44 1026.96 2.81 84/100 977.61 4.43 73/100 963.77 1.26 62/100 1063.16 17945.39 10 30 5 1213.86 1177.95 2.83 74/100 1119.21 4.67 73/100 1095.34 1.96 72/100 1215.71 24981.61 TableA.94: Complex-TaskNegotiations: ComparisonofTeamCostsinUnknownTerrain(DisjointCoalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 1 76.48 76.48 0.00 100/100 73.60 0.00 100/100 73.60 0.00 100/100 80.82 128.68 4 12 2 92.70 93.39 -1.53 72/100 81.83 1.45 85/100 81.83 1.45 85/100 96.63 208.44 6 18 3 95.26 93.16 1.23 66/100 86.82 0.98 61/100 86.82 0.98 61/100 97.75 242.06 8 24 4 97.18 95.40 1.11 60/100 85.36 2.10 73/100 85.36 2.10 73/100 96.04 253.79 10 30 5 96.57 92.94 2.53 59/100 85.15 1.59 61/100 85.15 1.59 61/100 94.77 264.29 MiniMaxTeamObjective 2 6 1 74.73 74.73 0.00 100/100 72.10 0.00 100/100 72.10 0.00 100/100 78.75 128.68 4 12 2 95.34 94.36 0.30 61/100 83.06 1.39 73/100 83.06 1.39 73/100 92.97 208.44 6 18 3 98.50 96.34 1.13 66/100 87.30 2.29 61/100 87.30 2.29 61/100 97.71 242.06 8 24 4 96.67 93.95 1.95 60/100 89.02 0.53 57/100 89.02 0.53 57/100 94.51 253.79 10 30 5 97.08 94.84 1.22 60/100 86.91 4.11 62/100 86.91 4.11 62/100 94.98 264.29 MiniLatTeamObjective 2 6 1 78.15 78.15 0.00 100/100 75.82 0.00 100/100 75.82 0.00 100/100 78.75 128.68 4 12 2 91.94 91.49 0.24 78/100 84.67 0.77 77/100 84.67 0.77 77/100 92.97 208.44 6 18 3 96.83 92.92 3.36 77/100 84.39 3.43 77/100 84.39 3.43 77/100 97.71 242.06 8 24 4 95.26 90.84 4.26 72/100 83.47 1.39 64/100 83.47 1.39 64/100 94.51 253.79 10 30 5 96.22 91.37 4.59 77/100 81.01 2.97 69/100 81.01 2.97 69/100 94.98 264.29 Table A.95: Complex-Task Negotiations: Comparison of Numbers of Repeated Executions in Unknown Terrain (Disjoint Coalitions) 384 Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 1 0.05 0.05 0.10 0.10 0.04 0.06 4 12 2 0.17 0.17 0.65 0.68 0.09 0.21 6 18 3 0.36 0.38 1.93 2.17 0.14 0.38 8 24 4 0.59 0.65 3.97 4.52 0.18 0.55 10 30 5 0.89 0.98 6.81 8.11 0.23 0.73 MiniMaxTeamObjective 2 6 1 0.05 0.05 0.11 0.11 0.04 0.06 4 12 2 0.17 0.17 0.75 0.88 0.09 0.21 6 18 3 0.34 0.36 2.30 2.49 0.14 0.38 8 24 4 0.56 0.63 4.49 4.68 0.19 0.55 10 30 5 0.83 0.93 6.61 7.23 0.25 0.73 MiniLatTeamObjective 2 6 1 0.05 0.05 0.10 0.10 0.04 0.06 4 12 2 0.17 0.18 0.60 0.71 0.09 0.21 6 18 3 0.33 0.36 1.76 1.96 0.15 0.38 8 24 4 0.55 0.63 3.73 4.70 0.19 0.55 10 30 5 0.79 0.92 6.46 8.09 0.24 0.72 TableA.96: Complex-TaskNegotiations: ComparisonofCumulativeResponseTimesinUnknownTerrain(DisjointCoalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 274.92 274.92 0.00 100/100 265.26 3.77 100/100 265.26 0.00 100/100 342.20 431.78 4 12 3 329.17 324.17 1.39 100/100 302.81 6.27 94/100 301.44 0.43 97/100 372.05 651.99 6 18 3 368.61 361.78 1.77 100/100 336.26 6.72 92/100 331.68 1.35 98/100 403.32 971.22 8 24 3 404.97 397.42 1.60 100/100 364.42 8.16 90/100 358.84 1.46 95/100 434.54 1084.38 10 30 3 446.55 438.11 1.78 100/100 396.19 9.34 93/100 388.66 1.81 99/100 474.34 1340.91 MiniMaxTeamObjective 2 6 3 143.75 143.75 0.00 100/100 137.98 4.36 100/100 137.98 0.00 100/100 158.21 230.87 4 12 3 113.43 101.73 9.19 100/100 94.42 6.19 78/100 91.69 2.48 93/100 120.22 202.24 6 18 3 104.26 91.16 11.77 99/100 77.01 13.47 84/100 74.06 3.06 86/100 110.90 236.80 8 24 3 100.43 89.54 10.24 100/100 67.53 22.03 93/100 65.09 3.03 90/100 105.26 190.51 10 30 3 97.43 88.41 8.49 100/100 63.29 26.49 96/100 61.55 1.89 83/100 101.39 217.21 MiniLatTeamObjective 2 6 3 600.77 600.77 0.00 100/100 578.83 3.67 100/100 578.83 0.00 100/100 616.69 1313.20 4 12 3 640.52 624.58 2.27 100/100 606.18 2.75 80/100 591.23 2.35 99/100 650.21 1695.85 6 18 3 710.31 684.11 3.55 100/100 650.91 4.56 82/100 634.94 2.33 90/100 716.98 2518.37 8 24 3 774.49 747.38 3.33 100/100 697.17 6.39 89/100 680.64 2.31 95/100 775.47 2563.85 10 30 3 848.30 827.07 2.40 100/100 754.47 8.41 99/100 743.34 1.38 89/100 855.40 3142.19 TableA.97: Complex-TaskNegotiations: ComparisonofTeamCostsinOutdoor-LikeTerrain(OverlappingCoalitions) 385 Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 3 0.01 0.01 0.13 0.13 0.00 0.00 4 12 3 0.01 0.01 0.29 0.29 0.00 0.00 6 18 3 0.01 0.01 0.25 0.26 0.00 0.00 8 24 3 0.01 0.01 0.32 0.33 0.00 0.00 10 30 3 0.01 0.02 0.40 0.43 0.00 0.00 MiniMaxTeamObjective 2 6 3 0.01 0.01 0.13 0.13 0.00 0.00 4 12 3 0.01 0.01 0.15 0.15 0.00 0.00 6 18 3 0.01 0.01 0.13 0.14 0.00 0.00 8 24 3 0.01 0.01 0.11 0.14 0.00 0.00 10 30 3 0.01 0.02 0.21 0.22 0.00 0.00 MiniLatTeamObjective 2 6 3 0.03 0.01 0.14 0.14 0.00 0.00 4 12 3 0.02 0.01 0.13 0.14 0.00 0.00 6 18 3 0.02 0.02 0.18 0.22 0.00 0.00 8 24 3 0.01 0.02 0.15 0.23 0.00 0.00 10 30 3 0.01 0.02 0.34 0.41 0.00 0.00 TableA.98: Complex-TaskNegotiations: ComparisonofResponseTimesinOutdoor-LikeTerrain(OverlappingCoalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 246.98 246.98 0.00 100/100 241.01 2.86 100/100 241.01 0.00 100/100 310.70 408.05 4 12 3 295.24 289.88 1.68 100/100 275.27 4.94 94/100 272.74 0.82 98/100 330.80 632.47 6 18 3 327.37 321.43 1.78 100/100 295.52 7.86 96/100 292.40 0.99 95/100 358.91 917.65 8 24 3 364.32 356.01 2.17 100/100 322.01 9.28 96/100 319.07 0.83 97/100 392.98 1040.07 10 30 3 398.26 390.54 1.81 100/100 347.14 10.89 96/100 341.44 1.54 97/100 419.66 1246.30 MiniMaxTeamObjective 2 6 3 129.95 129.95 0.00 100/100 125.00 4.74 100/100 125.00 0.00 100/100 144.96 219.52 4 12 3 100.23 91.72 7.67 100/100 84.46 7.22 80/100 81.95 2.58 94/100 107.12 197.95 6 18 3 94.23 81.35 11.89 100/100 68.27 14.09 87/100 66.30 2.54 93/100 96.90 226.52 8 24 3 89.99 80.69 9.42 100/100 61.36 21.91 94/100 59.33 2.73 93/100 93.58 184.95 10 30 3 88.35 81.15 7.21 100/100 57.18 27.19 96/100 56.07 1.26 82/100 91.07 191.41 MiniLatTeamObjective 2 6 3 545.97 545.97 0.00 100/100 532.85 2.59 100/100 532.85 0.00 100/100 562.87 1252.47 4 12 3 565.05 554.23 1.75 100/100 532.63 3.66 90/100 523.43 1.67 98/100 568.54 1657.58 6 18 3 627.39 610.62 2.57 100/100 577.64 5.13 84/100 562.48 2.47 93/100 630.03 2379.20 8 24 3 695.51 675.19 2.74 100/100 623.04 7.41 95/100 610.03 2.08 92/100 700.42 2455.84 10 30 3 757.80 736.45 2.70 100/100 668.70 8.94 97/100 658.48 1.47 90/100 761.24 2912.92 TableA.99: Complex-TaskNegotiations: ComparisonofTeamCostsinEmptyTerrain(OverlappingCoalitions) 386 Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 3 0.01 0.01 0.12 0.12 0.00 0.00 4 12 3 0.01 0.01 0.25 0.25 0.00 0.00 6 18 3 0.01 0.01 0.31 0.32 0.00 0.00 8 24 3 0.01 0.01 0.34 0.34 0.00 0.00 10 30 3 0.01 0.02 0.40 0.43 0.00 0.00 MiniMaxTeamObjective 2 6 3 0.01 0.01 0.13 0.13 0.00 0.00 4 12 3 0.01 0.01 0.16 0.17 0.00 0.00 6 18 3 0.01 0.01 0.17 0.17 0.00 0.00 8 24 3 0.01 0.01 0.10 0.13 0.00 0.00 10 30 3 0.01 0.02 0.21 0.22 0.00 0.00 MiniLatTeamObjective 2 6 3 0.03 0.03 0.12 0.12 0.00 0.00 4 12 3 0.02 0.02 0.14 0.18 0.00 0.00 6 18 3 0.02 0.02 0.17 0.22 0.00 0.00 8 24 3 0.01 0.02 0.11 0.17 0.00 0.00 10 30 3 0.01 0.02 0.34 0.41 0.00 0.00 TableA.100: Complex-TaskNegotiations: ComparisonofResponseTimesinEmptyTerrain(OverlappingCoalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 405.38 405.38 0.00 100/100 390.83 3.94 100/100 390.83 0.00 100/100 500.23 658.67 4 12 3 469.18 462.61 1.26 100/100 439.94 4.69 89/100 436.39 0.77 98/100 521.75 983.69 6 18 3 516.17 506.86 1.61 100/100 469.17 7.17 95/100 462.18 1.38 97/100 569.64 1481.25 8 24 3 558.38 545.56 2.17 100/100 497.87 8.48 95/100 488.75 1.70 98/100 604.52 1632.44 10 30 3 607.38 594.47 2.05 100/100 529.62 10.66 94/100 521.72 1.39 96/100 646.68 1967.98 MiniMaxTeamObjective 2 6 3 212.77 212.57 0.00 100/100 202.74 4.82 100/100 202.74 0.00 100/100 231.86 353.78 4 12 3 163.01 145.44 9.59 100/100 141.15 2.26 70/100 134.40 4.11 97/100 168.52 308.21 6 18 3 141.44 127.20 8.83 100/100 109.71 11.78 85/100 105.26 3.55 92/100 149.73 356.56 8 24 3 140.62 125.57 9.91 100/100 99.93 17.48 91/100 95.18 3.80 86/100 140.82 286.08 10 30 3 137.92 126.08 8.06 100/100 93.50 23.96 94/100 89.95 2.58 91/100 143.43 315.49 MiniLatTeamObjective 2 6 3 892.33 892.33 0.00 100/100 869.02 2.89 100/100 869.02 0.00 100/100 916.14 2026.69 4 12 3 907.20 884.58 2.32 100/100 864.17 2.13 82/100 846.77 1.83 98/100 913.17 2591.70 6 18 3 977.30 944.26 3.18 100/100 904.91 3.81 81/100 882.77 2.35 95/100 986.83 3867.27 8 24 3 1050.47 1019.29 2.78 100/100 966.89 4.90 88/100 936.87 2.90 96/100 1057.02 3855.45 10 30 3 1151.24 1116.73 2.87 100/100 1036.74 7.01 89/100 1011.79 2.27 91/100 1155.21 4650.90 TableA.101: Complex-TaskNegotiations: ComparisonofTeamCostsinIndoor-LikeTerrain(OverlappingCoalitions) 387 Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 3 0.01 0.01 0.14 0.15 0.00 0.00 4 12 3 0.01 0.01 0.27 0.28 0.00 0.00 6 18 3 0.01 0.01 0.26 0.28 0.00 0.00 8 24 3 0.01 0.01 0.27 0.30 0.00 0.00 10 30 3 0.01 0.02 0.41 0.44 0.00 0.00 MiniMaxTeamObjective 2 6 3 0.01 0.01 0.14 0.14 0.00 0.00 4 12 3 0.01 0.01 0.19 0.17 0.00 0.00 6 18 3 0.01 0.01 0.13 0.15 0.00 0.00 8 24 3 0.01 0.01 0.15 0.18 0.00 0.00 10 30 3 0.01 0.02 0.21 0.22 0.00 0.00 MiniLatTeamObjective 2 6 3 0.03 0.03 0.13 0.13 0.00 0.00 4 12 3 0.02 0.02 0.18 0.19 0.00 0.00 6 18 3 0.02 0.02 0.11 0.15 0.00 0.00 8 24 3 0.01 0.02 0.13 0.19 0.00 0.00 10 30 3 0.01 0.02 0.34 0.41 0.00 0.00 TableA.102: Complex-TaskNegotiations: ComparisonofResponseTimesinIndoor-LikeTerrain(OverlappingCoalitions) Simple- ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Target SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets Capacities (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 3 2 373.86 366.19 1.89 100/100 334.24 8.41 93/100 327.70 1.86 94/100 397.18 975.86 10 30 3 3 446.55 438.11 1.78 100/100 396.19 9.34 93/100 388.66 1.81 99/100 474.34 1340.91 10 40 3 4 494.36 486.14 1.57 100/100 444.44 8.37 95/100 438.93 1.18 96/100 532.16 1692.78 10 50 3 5 542.14 532.90 1.64 100/100 492.57 7.47 93/100 486.29 1.20 97/100 582.91 2008.83 MiniMaxTeamObjective 10 20 3 2 82.06 74.00 9.27 100/100 55.17 21.94 85/100 52.67 3.42 90/100 84.04 162.83 10 30 3 3 97.43 88.41 8.49 100/100 63.29 26.49 96/100 61.55 1.89 83/100 101.39 217.21 10 40 3 4 107.87 96.56 9.90 100/100 70.80 24.25 91/100 67.92 3.02 82/100 110.81 253.02 10 50 3 5 111.41 102.90 6.90 100/100 77.31 23.38 94/100 75.83 1.50 79/100 121.10 287.84 MiniLatTeamObjective 10 20 3 2 555.10 539.09 2.61 100/100 497.87 7.46 93/100 487.62 1.84 94/100 556.51 1774.52 10 30 3 3 848.30 827.07 2.40 100/100 754.47 8.41 99/100 743.34 1.38 89/100 855.40 3142.19 10 40 3 4 1177.79 1142.40 2.84 100/100 1038.37 8.91 95/100 1018.22 1.82 92/100 1187.15 4832.00 10 50 3 5 1547.15 1510.88 2.26 100/100 1365.91 9.38 99/100 1342.94 1.61 92/100 1565.89 6790.31 TableA.103: Complex-TaskNegotiations: ComparisonofTeamCostsforDifferentRobotSimple-TargetCapacities(Overlap- pingCoalitions) 388 Robots SimpleTargets ComplexTargets Simple-TargetCapacities SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 20 3 2 0.00 0.00 0.03 0.04 0.00 0.00 10 30 3 3 0.01 0.02 0.40 0.43 0.00 0.00 10 40 3 4 0.01 0.06 2.36 2.47 0.01 0.00 10 50 3 5 0.04 0.13 7.66 8.49 0.01 0.00 MiniMaxTeamObjective 10 20 3 2 0.00 0.00 0.01 0.02 0.00 0.00 10 30 3 3 0.01 0.02 0.21 0.22 0.00 0.00 10 40 3 4 0.01 0.05 0.89 0.99 0.00 0.00 10 50 3 5 0.04 0.14 5.68 6.67 0.01 0.00 MiniLatTeamObjective 10 20 3 2 0.00 0.00 0.01 0.03 0.00 0.00 10 30 3 3 0.01 0.02 0.34 0.41 0.00 0.00 10 40 3 4 0.02 0.07 1.45 1.65 0.01 0.00 10 50 3 5 0.05 0.20 6.15 7.93 0.01 0.00 TableA.104: Complex-TaskNegotiations: ComparisonofResponseTimesforDifferentRobotSimple-TargetCapacities(Over- lappingCoalitions) Simple- ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Target SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets Capacities (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 20 3 2 373.25 364.81 2.10 100/100 333.74 8.18 93/100 327.18 1.87 96/100 397.18 975.86 10 30 3 3 445.98 437.37 1.84 100/100 394.59 9.53 93/100 387.27 1.75 99/100 474.34 1340.91 10 40 3 4 493.00 484.53 1.63 100/100 442.47 8.49 96/100 437.59 1.05 93/100 532.16 1692.78 10 50 3 5 540.58 530.07 1.86 100/100 489.70 7.54 94/100 483.31 1.21 96/100 582.91 2008.83 MiniMaxTeamObjective 10 20 3 2 81.62 74.08 8.62 100/100 54.81 22.60 87/100 52.75 2.86 88/100 84.04 162.83 10 30 3 3 97.45 87.75 9.20 100/100 63.63 25.36 94/100 61.09 2.76 82/100 101.39 217.21 10 40 3 4 107.33 97.17 9.11 100/100 70.13 25.27 90/100 67.24 3.23 88/100 110.81 253.02 10 50 3 5 110.78 104.05 5.88 100/100 77.61 23.65 94/100 75.70 1.75 81/100 121.10 287.84 MiniLatTeamObjective 10 20 3 2 554.31 537.65 2.79 100/100 497.41 7.31 92/100 486.88 1.90 93/100 556.51 1774.52 10 30 3 3 848.59 825.92 2.56 100/100 753.43 8.43 99/100 741.08 1.53 93/100 855.40 3142.19 10 40 3 4 1173.66 1141.40 2.62 100/100 1032.42 9.31 97/100 1015.54 1.56 91/100 1187.15 4832.00 10 50 3 5 1534.65 1504.43 1.91 100/100 1361.07 9.35 100/100 1337.85 1.63 90/100 1565.89 6790.31 Table A.105: Complex-Task Negotiations: Comparison of Team Costs for Ideal Reaction-Function Calculations (Overlapping Coalitions) 389 Robots SimpleTargets ComplexTargets Simple-TargetCapacities SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 20 3 2 0.00 0.00 0.03 0.04 0.00 0.00 10 30 3 3 0.01 0.02 0.53 0.74 0.00 0.00 10 40 3 4 0.01 0.09 2.86 3.62 0.01 0.00 10 50 3 5 0.05 0.21 8.44 9.92 0.01 0.00 MiniMaxTeamObjective 10 20 3 2 0.00 0.00 0.01 0.02 0.00 0.00 10 30 3 3 0.01 0.02 0.34 0.42 0.00 0.00 10 40 3 4 0.01 0.07 1.28 1.52 0.00 0.00 10 50 3 5 0.05 0.24 5.84 7.02 0.01 0.00 MiniLatTeamObjective 10 20 3 2 0.00 0.00 0.01 0.03 0.00 0.00 10 30 3 3 0.02 0.03 0.55 0.69 0.00 0.00 10 40 3 4 0.03 0.09 1.79 2.00 0.01 0.00 10 50 3 5 0.08 0.31 8.35 9.48 0.01 0.00 TableA.106: Complex-TaskNegotiations: ComparisonofResponseTimesforIdealReaction-FunctionCalculations(Overlap- pingCoalitions) Simple- ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Target SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets Capacities (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 30 3 3 445.62 437.76 1.68 100/100 409.31 6.35 92/100 400.21 1.03 52/100 474.34 1340.91 30 3 4 363.49 360.39 0.79 100/100 338.81 5.72 95/100 333.77 0.62 63/100 394.38 1301.69 30 3 5 338.44 336.72 0.48 100/100 322.18 4.15 96/100 321.65 0.16 98/100 369.09 1316.08 30 3 6 328.50 327.57 0.27 100/100 315.46 3.62 98/100 315.68 -0.07 99/100 358.76 1316.08 30 3 7 324.97 324.31 0.19 100/100 311.61 3.85 98/100 310.81 0.24 98/100 354.08 1316.08 30 3 ∞ 314.90 314.41 0.15 100/100 305.94 2.67 97/100 305.54 0.13 99/100 345.48 1316.08 MiniMaxTeamObjective 30 3 3 97.26 87.68 9.10 100/100 69.25 18.80 90/100 60.61 9.85 60/100 101.39 217.21 30 3 4 62.33 60.61 2.50 100/100 51.77 13.46 97/100 52.85 -1.94 77/100 63.67 229.03 30 3 5 58.04 57.10 1.58 100/100 50.33 10.75 92/100 51.02 -1.30 95/100 58.25 268.10 30 3 6 56.14 55.00 1.88 100/100 50.14 8.17 87/100 49.65 0.92 98/100 56.87 268.10 30 3 7 55.78 54.90 1.49 100/100 50.04 8.23 89/100 49.16 1.57 98/100 56.21 268.10 30 3 ∞ 55.51 54.58 1.57 100/100 50.18 7.56 89/100 49.38 1.43 99/100 56.02 268.10 MiniLatTeamObjective 30 3 3 848.56 826.04 2.55 100/100 785.03 4.82 91/100 805.60 -2.56 84/100 855.40 3142.19 30 3 4 716.46 712.25 0.58 100/100 681.02 4.13 94/100 681.52 -0.08 93/100 720.17 3438.95 30 3 5 693.69 691.38 0.33 100/100 665.78 3.47 95/100 661.51 0.61 97/100 695.89 3647.22 30 3 6 687.59 686.23 0.20 100/100 663.73 3.08 96/100 659.73 0.57 98/100 689.82 3647.22 30 3 7 686.39 685.01 0.20 100/100 662.69 3.05 95/100 660.16 0.35 97/100 688.35 3647.22 30 3 ∞ 686.12 684.65 0.21 100/100 662.70 3.01 95/100 659.17 0.50 98/100 688.14 3647.22 Table A.107: Complex-Task Negotiations: Comparison of Team Costs for Loose Robot Simple-Target Capacity Constraints (OverlappingCoalitions) 390 Robots SimpleTargets ComplexTargets Simple-TargetCapacities SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 30 3 3 0.01 0.02 0.40 0.43 0.00 0.00 10 30 3 4 0.01 0.03 0.58 0.66 0.00 0.00 10 30 3 5 0.02 0.05 0.88 1.03 0.00 0.00 10 30 3 6 0.04 0.08 2.47 2.66 0.00 0.00 10 30 3 7 0.06 0.12 3.50 3.58 0.00 0.00 10 30 3 ∞ 0.33 0.60 20.35 21.47 0.00 0.00 MiniMaxTeamObjective 10 30 3 3 0.01 0.02 0.21 0.22 0.00 0.00 10 30 3 4 0.01 0.02 0.16 0.19 0.00 0.00 10 30 3 5 0.01 0.02 0.14 0.20 0.00 0.00 10 30 3 6 0.01 0.02 0.15 0.20 0.00 0.00 10 30 3 7 0.01 0.02 0.14 0.18 0.00 0.00 10 30 3 ∞ 0.01 0.02 0.14 0.18 0.00 0.00 MiniLatTeamObjective 10 30 3 3 0.01 0.02 0.34 0.41 0.00 0.00 10 30 3 4 0.01 0.02 0.18 0.30 0.00 0.00 10 30 3 5 0.01 0.02 0.21 0.28 0.00 0.00 10 30 3 6 0.01 0.02 0.20 0.28 0.00 0.00 10 30 3 7 0.01 0.02 0.21 0.28 0.00 0.00 10 30 3 ∞ 0.01 0.02 0.21 0.27 0.00 0.00 Table A.108: Complex-Task Negotiations: Comparison of Response Times for Loose Robot Simple-Target Capacity Con- straints(OverlappingCoalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 10 30 3 446.55 438.11 1.78 100/100 396.19 9.34 93/100 388.66 1.81 99/100 474.34 1340.91 10 30 4 471.56 463.22 0.84 100/100 422.25 8.76 96/100 413.99 1.87 98/100 504.24 1395.93 10 30 5 491.10 484.94 0.46 100/100 443.91 8.34 95/100 438.00 1.26 95/100 530.83 1539.48 10 30 6 520.74 503.22 2.40 100/100 461.52 8.08 94/100 455.08 1.32 95/100 560.95 1716.98 10 30 7 557.94 532.53 3.89 100/100 492.23 7.40 90/100 483.77 1.63 90/100 595.51 1895.78 MiniMaxTeamObjective 10 30 3 97.43 88.41 8.49 100/100 63.29 26.49 96/100 61.55 1.89 83/100 101.39 217.21 10 30 4 100.60 88.45 7.84 100/100 65.61 23.80 93/100 63.96 1.96 83/100 103.46 217.69 10 30 5 101.08 88.02 8.82 100/100 70.28 18.20 91/100 68.46 1.95 81/100 108.13 244.05 10 30 6 110.95 92.48 13.12 100/100 71.11 21.08 94/100 69.21 1.82 77/100 114.17 279.80 10 30 7 107.16 90.85 11.81 100/100 75.26 15.21 84/100 72.95 2.28 80/100 113.42 314.34 MiniLatTeamObjective 10 30 3 848.30 827.07 2.40 100/100 754.47 8.41 99/100 743.34 1.38 89/100 855.40 3142.19 10 30 4 902.24 877.97 0.05 100/100 819.00 6.45 93/100 797.41 2.55 97/100 910.92 3360.11 10 30 5 959.78 936.33 0.58 100/100 879.23 5.92 93/100 851.98 2.99 94/100 975.78 3918.82 10 30 6 1027.82 989.16 1.39 100/100 935.08 5.22 90/100 910.20 2.57 96/100 1031.48 4610.66 10 30 7 1083.62 1043.37 3.60 100/100 996.07 4.35 89/100 966.64 2.87 98/100 1088.15 5407.00 TableA.109: Complex-TaskNegotiations: ComparisonofTeamCostsforDifferentNumberofComplexTargets(Overlapping Coalitions) 391 Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 10 30 3 0.01 0.02 0.40 0.43 0.00 0.00 10 30 4 0.01 0.05 0.79 0.90 0.00 0.00 10 30 5 0.03 0.21 1.46 2.09 0.00 0.00 10 30 6 0.06 2.76 2.82 9.77 0.00 0.00 10 30 7 0.66 43.73 31.91 104.75 0.00 0.00 MiniMaxTeamObjective 10 30 3 0.01 0.02 0.21 0.22 0.00 0.00 10 30 4 0.01 0.04 0.24 0.33 0.00 0.00 10 30 5 0.02 0.10 0.69 0.84 0.00 0.00 10 30 6 0.03 0.36 1.40 1.84 0.00 0.00 10 30 7 0.08 4.78 3.40 7.76 0.00 0.00 MiniLatTeamObjective 10 30 3 0.01 0.02 0.34 0.41 0.00 0.00 10 30 4 0.01 0.05 0.44 0.50 0.00 0.00 10 30 5 0.02 0.33 1.05 1.87 0.00 0.00 10 30 6 0.11 6.64 2.82 23.26 0.00 0.00 10 30 7 2.94 151.97 77.52 367.58 0.00 0.00 Table A.110: Complex-Task Negotiations: Comparison of Response Times for Different Number of Complex Targets (Over- lappingCoalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex Random SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions SARF Targets Targets (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance (Mixed) MiniSumTeamObjective 2 6 3 431.78 289.14 32.10 100/100 265.53 7.97 100/100 265.53 0.00 100/100 342.20 274.92 4 12 3 651.99 427.21 33.97 100/100 353.51 16.89 97/100 333.79 5.26 93/100 372.05 329.17 6 18 3 971.22 567.12 41.12 100/100 432.76 23.37 100/100 376.81 12.38 97/100 403.32 368.61 8 24 3 1084.38 704.10 34.80 100/100 476.91 31.87 100/100 420.76 11.46 93/100 434.54 404.97 10 30 3 1340.91 871.93 34.79 100/100 554.52 36.17 100/100 475.96 13.44 93/100 474.34 446.55 MiniMaxTeamObjective 2 6 3 230.87 151.47 33.21 100/100 138.09 8.61 100/100 138.09 0.00 100/100 158.21 143.75 4 12 3 202.24 123.00 38.34 100/100 104.09 14.93 91/100 99.65 3.73 90/100 120.22 113.43 6 18 3 236.80 116.65 49.60 100/100 102.76 11.26 84/100 90.23 10.96 88/100 110.90 104.26 8 24 3 190.51 112.31 40.22 100/100 90.35 18.93 95/100 87.23 2.84 72/100 105.26 100.43 10 30 3 217.21 116.23 45.54 100/100 97.31 15.53 89/100 89.16 6.82 75/100 101.39 97.43 MiniLatTeamObjective 2 6 3 1313.20 682.49 46.56 100/100 582.85 14.36 100/100 582.85 0.00 100/100 616.69 600.77 4 12 3 1695.85 904.59 45.82 100/100 648.46 27.88 100/100 618.65 4.44 93/100 650.21 640.52 6 18 3 2518.37 1177.45 52.41 100/100 736.81 37.16 100/100 666.13 9.30 93/100 716.98 710.31 8 24 3 2563.85 1424.14 44.06 100/100 777.52 45.05 100/100 716.19 7.68 94/100 775.47 774.49 10 30 3 3142.19 1739.53 44.30 100/100 873.21 49.63 100/100 790.37 9.43 96/100 855.40 848.30 TableA.111: Complex-TaskNegotiations: ComparisonofTeamCostsforRandomizedInitialAllocations(OverlappingCoali- tions) 392 Robots SimpleTargets ComplexTargets Random SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions SARF(Mixed) MiniSumTeamObjective 2 6 3 0.00 0.01 0.16 0.16 0.00 0.01 4 12 3 0.00 0.01 0.27 0.30 0.00 0.01 6 18 3 0.00 0.02 0.33 0.41 0.00 0.01 8 24 3 0.00 0.03 0.47 0.51 0.00 0.01 10 30 3 0.00 0.03 0.60 0.68 0.00 0.01 MiniMaxTeamObjective 2 6 3 0.00 0.01 0.16 0.16 0.00 0.01 4 12 3 0.00 0.01 0.19 0.21 0.00 0.01 6 18 3 0.00 0.02 0.25 0.29 0.00 0.01 8 24 3 0.00 0.02 0.34 0.39 0.00 0.01 10 30 3 0.00 0.02 0.48 0.51 0.00 0.01 MiniLatTeamObjective 2 6 3 0.00 0.01 0.21 0.20 0.00 0.03 4 12 3 0.00 0.01 0.24 0.28 0.00 0.02 6 18 3 0.00 0.03 0.30 0.41 0.00 0.02 8 24 3 0.00 0.04 0.38 0.52 0.00 0.01 10 30 3 0.00 0.05 0.53 0.70 0.00 0.01 Table A.112: Complex-Task Negotiations: Comparison of Response Times for Randomized Initial Allocations (Overlapping Coalitions) ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 341.36 341.36 0.00 100/100 338.95 2.06 76/100 338.95 0.00 100/100 419.84 1635.56 4 12 3 413.48 409.42 0.50 61/100 401.75 1.55 64/100 398.50 0.50 76/100 462.12 3652.85 6 18 3 462.67 456.01 1.20 66/100 433.95 4.39 71/100 431.38 0.28 68/100 499.51 5883.31 8 24 3 498.81 496.51 0.07 61/100 467.24 5.35 75/100 459.41 1.30 71/100 530.29 7981.37 10 30 3 543.27 542.65 -0.72 54/100 504.03 6.99 78/100 501.17 0.33 70/100 568.85 9889.90 MiniMaxTeamObjective 2 6 3 184.53 184.53 0.00 100/100 176.81 4.17 80/100 176.81 0.00 100/100 198.83 917.31 4 12 3 150.15 145.15 1.48 64/100 130.63 6.97 68/100 125.50 2.45 66/100 150.63 1162.61 6 18 3 137.62 132.84 0.41 63/100 110.73 13.71 80/100 108.66 0.22 52/100 140.15 1340.92 8 24 3 135.42 125.13 5.30 70/100 101.83 14.15 74/100 99.01 1.14 61/100 127.19 1447.78 10 30 3 124.76 121.52 0.15 66/100 96.89 16.83 84/100 89.90 4.94 70/100 123.78 1465.01 MiniLatTeamObjective 2 6 3 753.72 753.72 0.00 100/100 739.85 2.27 83/100 739.85 0.00 100/100 790.99 4911.17 4 12 3 787.01 770.93 1.79 79/100 748.07 2.59 70/100 737.92 1.24 75/100 797.92 9289.97 6 18 3 865.24 838.88 2.85 79/100 817.10 2.17 62/100 796.47 1.93 71/100 891.53 14114.89 8 24 3 931.95 905.40 2.52 83/100 853.13 5.35 77/100 837.89 1.50 75/100 946.23 18269.26 10 30 3 1026.24 1000.43 2.21 76/100 931.74 6.51 85/100 923.89 0.72 67/100 1043.57 22392.65 TableA.113: Complex-TaskNegotiations: ComparisonofTeamCostsinUnknownTerrain(OverlappingCoalitions) 393 ComparisonofSARF(Mixed) ComparisonofSNRF(Complex- ComparisonofSNRF(Simple- Robots Simple Complex SARF SNRF andSNRF(Complex-Only) SNRF Only)andSNRF(Simple-Only) SNRF Only)andSNRF(Mixed) SKAuctions Random Targets Targets (Mixed) (Complex-Only) Difference(%) Dominance (Simple-Only) Difference(%) Dominance (Mixed) Difference(%) Dominance MiniSumTeamObjective 2 6 3 90.81 90.81 0.00 100/100 89.54 1.49 72/100 89.54 0.00 100/100 99.92 154.10 4 12 3 96.34 93.75 1.93 71/100 89.04 3.99 75/100 89.50 -1.03 71/100 99.05 223.14 6 18 3 90.01 88.61 1.05 67/100 84.01 4.85 69/100 84.13 -0.77 62/100 90.52 249.32 8 24 3 87.11 87.76 -1.21 71/100 78.30 9.71 81/100 76.66 1.43 72/100 85.49 257.28 10 30 3 83.67 84.40 -1.57 63/100 73.66 11.57 78/100 72.94 0.34 64/100 81.66 263.94 MiniMaxTeamObjective 2 6 3 91.96 91.83 0.00 100/100 88.59 2.32 75/100 88.59 0.00 100/100 95.48 154.10 4 12 3 96.24 97.13 -1.90 54/100 90.08 5.95 67/100 89.92 -0.73 54/100 95.39 223.14 6 18 3 93.37 94.05 -2.24 55/100 85.10 8.21 79/100 84.43 0.00 53/100 93.36 249.32 8 24 3 91.32 88.90 1.74 67/100 80.32 8.15 71/100 80.56 -1.12 52/100 88.66 257.28 10 30 3 86.24 86.66 -1.37 57/100 78.15 8.29 71/100 73.99 4.42 70/100 85.40 263.94 MiniLatTeamObjective 2 6 3 93.55 93.55 0.00 100/100 92.40 1.74 81/100 92.40 0.00 100/100 95.48 154.10 4 12 3 95.04 92.59 2.05 75/100 89.36 3.20 65/100 87.69 1.53 75/100 95.39 223.14 6 18 3 92.16 87.80 4.14 79/100 82.12 5.35 68/100 80.01 1.83 71/100 93.36 249.32 8 24 3 88.44 85.46 2.81 76/100 72.54 13.52 82/100 71.90 0.10 65/100 88.66 257.28 10 30 3 84.28 81.76 2.35 74/100 69.65 13.83 90/100 68.67 0.88 71/100 85.40 263.94 TableA.114: Complex-TaskNegotiations: ComparisonofNumbersofRepeatedExecutionsinUnknownTerrain(Overlapping Coalitions) Robots SimpleTargets ComplexTargets SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) SKAuctions Random MiniSumTeamObjective 2 6 3 0.65 0.74 1.97 2.89 0.04 0.10 4 12 3 0.67 0.71 3.54 4.55 0.09 0.25 6 18 3 0.68 0.73 3.48 4.77 0.15 0.41 8 24 3 0.70 0.86 2.98 3.97 0.19 0.55 10 30 3 0.71 0.87 2.47 3.84 0.24 0.70 MiniMaxTeamObjective 2 6 3 0.75 0.75 2.04 3.04 0.04 0.11 4 12 3 0.68 0.73 2.20 3.17 0.09 0.26 6 18 3 0.67 0.74 1.45 2.54 0.15 0.43 8 24 3 0.71 0.86 1.47 2.63 0.19 0.62 10 30 3 0.72 0.77 1.25 2.50 0.22 0.78 MiniLatTeamObjective 2 6 3 0.75 0.76 1.63 2.69 0.05 0.10 4 12 3 0.74 0.79 1.37 2.71 0.10 0.25 6 18 3 0.73 0.74 1.09 2.48 0.16 0.40 8 24 3 0.76 0.79 1.26 2.76 0.18 0.56 10 30 3 0.78 0.80 1.28 2.94 0.23 0.75 Table A.115: Complex-Task Negotiations: Comparison of Cumulative Response Times in Unknown Terrain (Overlapping Coalitions) 394 Robots Simple Complex SARF(Complex-Only) SARF(Simple-Only) SARF(Mixed) Targets Targets Profitable Negotiation Profitable Negotiation Profitable Negotiation TaskExchanges Rounds TaskExchanges Rounds TaskExchanges Rounds MiniSumTeamObjective 2 6 1 0.00 1.00 1.44 1.58 1.44 1.58 4 12 2 0.65 1.42 5.74 2.88 6.39 3.07 6 18 3 1.63 1.87 10.42 4.11 12.05 4.41 8 24 4 2.86 2.42 14.49 4.84 17.35 5.71 10 30 5 2.88 2.63 18.12 6.31 21.00 6.69 MiniMaxTeamObjective 2 6 1 0.00 1.00 2.78 1.71 2.78 1.71 4 12 2 0.86 1.46 10.30 2.99 11.16 3.10 6 18 3 1.71 1.92 16.26 3.87 17.97 4.07 8 24 4 3.63 2.24 23.31 5.08 26.94 4.94 10 30 5 4.61 2.54 35.08 5.44 39.69 5.82 MiniLatTeamObjective 2 6 1 0.00 1.00 1.38 1.57 1.38 1.57 4 12 2 0.70 1.47 4.40 2.85 5.10 3.06 6 18 3 1.47 1.83 7.65 3.93 9.12 4.50 8 24 4 1.91 2.17 11.21 5.66 13.12 6.40 10 30 5 2.82 2.47 15.82 6.88 18.64 8.43 Table A.116: Comparison of Numbers of Profitable Task Exchanges and Negotiation RoundsforInitialAllocationsProducedbySARF(Mixed)Algorithm Table A.80 tabulates the team costs of our task-allocation algorithms for disjoint coalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • The team costs of the SNRF (Complex-Only) algorithm are identical to those of the SARF (Mixed) algorithm (that produces the initial allocations), and the team costsoftheSNRF(Mixed)algorithmareidenticaltothoseoftheSNRF(Simple- Only) algorithm when the number of complex targets is one (no matter what the team objective is). This observation meets our expectation since no complex- target exchanges are performed when there are only two robots and one complex target. • TheteamcostsoftheSNRF(Complex-Only)algorithmaresmallerthanthoseof the SARF (Mixed) algorithm when the the number of complex targets is larger than one (no matter what the team objective is). For the MiniSum team objec- tive, the SNRF (Complex-Only) algorithm reduces the team costs of the SARF (Mixed) on average algorithm by 4.07 percent. For the MiniMax team objective, theSNRF(Complex-Only)algorithmreducestheteamcostsoftheSARF(Mixed) algorithmonaverageby9.84percent. FortheMiniLatteamobjective,theSNRF (Complex-Only)algorithmreducestheteamcostsoftheSARF(Mixed)algorithm 395 on average by 2.05 percent. This observation meets our expectation since the SNRF (Complex-Only) algorithm repeatedly constructs and performs profitable complex-targetexchangesbetweentworobotstoimprovethecurrentallocations. It is guaranteed that the SNRF (Complex-Only) algorithm does not increase the teamcostsoftheinitialallocationsproducedbytheSARF(Mixed)algorithm. • TheteamcostsoftheSNRF(Simple-Only)algorithmaresmallerthanthoseofthe SNRF (Complex-Only) algorithm (no matter what the team objective is). For the MiniSum team objective, the SNRF (Simple-Only) algorithm reduces the team costs of SNRF (Complex-Only) algorithm on average by 5.77 percent. For the MiniMax team objective, the SNRF (Simple-Only) algorithm reduces the team costs of the SNRF (Complex-Only) algorithm by on average 14.17 percent. For theMiniLatteamobjective,theSNRF(Simple-Only)algorithmreducestheteam costs of the SNRF (Complex-Only) algorithm on average by 6.16 percent. One- sided binomial tests [CB90] with normal approximation (n = 100; p = 0.5; σ 2 = 50)areperformedtosupportthehypothesisthattheteamcostsoftheSNRF (Simple-Only) algorithm are not larger than those of the SNRF (Complex-Only) algorithm with confidence 0.9995 (which is significant). This observation meets our expectation since SNRF (Simple-Only) algorithm repeatedly constructs and performssimple-targetexchangesbetweentworobotstoimprovethecurrentallo- cationsandtherearemanymoreprofitablesimple-targetexchangesthancomplex- targetexchanges. Toimproveourunderstandingofthisexplanation,wetabulates thefollowingadditionalresultsfortheSNRFalgorithmsinTableA.116: – Profitable Task Exchanges: The number of profitable task exchanges that constructed by robots that use the SNRF algorithms on the initial alloca- tions. 396 – NegotiationRounds: Thenumberoftaskexchangesrobotstotallyperform toimprovethecurrentallocationsbeforetheSNRFalgorithmsterminate. Theresultsdemonstratethatthenumberoftheprofitablesimple-targetexchanges areindeedlargerthanthatoftheprofitablecomplex-targetexchanges,androbots that use the SNRF (Simple-Only) algorithm performs more task exchanges than robotsthatusetheSNRF(Complex-Only)algorithm. • TheteamcostsoftheSNRF(Mixed)algorithmaresmallerthanthoseoftheSNRF (Simple-Only)algorithmwhenthenumberofcomplextaregetsislargerthanone (nomatterwhattheteamobjectiveis). FortheMiniSumteamobjective,theSNRF (Mixed) algorithm reduces the team costs of the SNRF (Simple-Only) algorithm by on average 2.29 percent. For the MiniMax team objective, the SNRF (Mixed) algorithm reduces the team costs of the SNRF (Simple-Only) algorithm on aver- ageby2.87percent. FortheMiniLatteamobjective,theSNRF(Mixed)algorithm reduces the team costs of the SNRF (Simple-Only) algorithm on average by 1.51 percent. Statistical tests support the hypothesis that the team costs of the SNRF (Mixed)algorithmarenotlargerthanthoseoftheSNRF(Simple-Only)algorithm with confidence 0.9995 (which is significant). This observation meets our expec- tationsincetheSNRF(Mixed)algorithmrepeatedlyconstructsandperformsboth simple-target and complex-target exchanges between two robots to improve the currentallocations. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithmsaresmallerthanthoseofSKAuctionsandtherandomizedalgorithm. This observationmeetsourexpectationsincetheteamcostsoftheSARF(Mixed)algo- rithmarealreadysmallerthanthoseofSKAuctionsandtherandomizedalgorithm and it is guaranteed that the SNRF (Complex-Only, Simple-Only, and Mixed) 397 algorithms do not increase the team costs of the initial allocations produced by theSARF(Mixed)algorithm. Table A.81 tabulates the response times of our task-allocation algorithms for dis- jointcoalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • Theresponsetimesofourtask-allocationalgorithmsareallsmall(nomatterwhat the team objective is). This observation demonstrates that the SNRF (Mixed) algorithmre-allocatestargetsamongrobotsefficiently. • TheresponsetimesoftheSNRF(Simple-Only)algorithmarelargerthanthoseof theSNRF(Complex-Only)algorithm(nomatterwhattheteamobjectiveis). This observation meets our expectation since: i) robots that use the SNRF (Complex- Only)algorithmconstructtheirreactionfunctionsforeachcomplextargetexactly once but robots that use the SNRF (Simple-Only) algorithm have to update their reaction functions for each simple-target exchange; and ii) there are many more simple-targetexchangesthancomplex-targetexchangesinourexperiments. • The response times of the SNRF (Mixed) algorithm are larger than those of the SNRF (Simple-Only) algorithm (no matter what the team objective is). This observation meets our expectation since robots that use the SNRF (Mixed) algo- rithmneedtoevaluatebothsimple-targetandcomplex-targetexchanges. Tables A.82 and A.84 tabulate the team costs of our task-allocation algorithms for disjoint coalitions inempty andindoor-like terrain, respectively. We make the follow- ingobservations: • The team costs of our task-allocation algorithms in empty terrain are all smaller than those in outdoor-like terrain and the team costs of our task-allocation algo- rithms in indoor-like terrain are all larger than those in outdoor-like terrain (no 398 matter what the team objective is). This observation meets our expectation since robotsinemptyterraindonothavetocircumnavigateanyblockedcellsandrobots in indoor-like terrain have to go through doors at fixed positions to visit targets, whereas they only have to circumnavigate groups of randomly blocked cells in outdoor-liketerrain. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithmsareallsmallerthanthoseoftheSARF(Mixed)algorithminbothemptyand indoor-like terrain (no matter what the team objective is). These results demon- strate that the SNRF (Mixed) algorithm reduces the team costs of the initial allo- cationsproducedbytheSARF(Mixed)algorithmeffectivelyindifferentkindsof terrain. TablesA.83andA.85tabulatetheresponsetimesofourtask-allocationalgorithms with disjoint coalitions in empty and indoor-like terrain, respectively. The results are almost identical to those in outdoor-like terrain shown in Table A.81 and thus support our previous statement that SNRF (Mixed) algorithm re-allocates targets among robots efficientlyindifferentkindsofterrain. Table A.86 tabulates the team costs of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain with different robot simple-target capacities. We makethefollowingobservations: • The team costs of our task-allocation algorithms all increase when the robot simple-target capacity increases (no matter what the team objective is). This observation meets our expectation since robots have to visit more simple targets whenthenumberofsimpletargetsisincreased. 399 Robots Simple Complex Simple-Target SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed)) Targets Targets Capacities TeamCost ResponseTime TeamCost ResponseTime TeamCost ResponseTime MiniSumTeamObjective 10 20 5 2 490.35 0.00 462.57 0.01 446.82 0.01 10 30 5 3 555.01 0.01 517.66 0.10 504.12 0.11 10 40 5 4 601.05 0.06 566.80 0.88 547.12 1.06 10 50 5 5 657.66 0.20 610.18 4.46 596.13 4.73 MiniMaxTeamObjective 10 20 5 2 80.56 0.00 68.20 0.01 65.43 0.01 10 30 5 3 94.74 0.01 76.30 0.09 72.93 0.10 10 40 5 4 103.82 0.05 83.30 0.77 80.29 0.84 10 50 5 5 112.69 0.18 90.07 3.40 88.23 3.30 MiniLatTeamObjective 10 20 5 2 673.27 0.00 633.98 0.01 616.79 0.01 10 30 5 3 980.69 0.01 914.90 0.12 898.05 0.14 10 40 5 4 1322.09 0.06 1211.22 1.15 1191.71 1.35 10 50 5 5 1697.93 0.19 1558.27 6.02 1532.17 6.81 TableA.117: TeamCostsandResponseTimesforSNRFAlgorithmswithDiscretization Granularity10(DisjointCoalitions) • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms are all smaller than those of the SARF (Mixed) for different robot simple- targetcapacities(nomatterwhattheteamobjectiveis). Theseresultsdemonstrate thatthatSNRF(Mixed)algorithmreducestheteamcostsoftheinitialallocations produced by the SARF (Mixed) algorithm effectively for different robot simple- targetcapacities. Table A.87 tabulates the response times of our task-allocation algorithms for dis- joint coalitions in outdoor-like terrain with different robot simple-target capacities. Wemakethefollowingobservations: • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms increase when the robot simple-target capacity increases (no matter what theteamobjectiveis). Thisobservationmeetsourexpectationsinceittakesmore time for robots to construct their reaction functions and the auctioneer to deter- minethevisittimesofcomplextargetswhentherobotsneedtovisitmoretargets. • The response times of the SNRF (Simple-Only and Mixed) algorithms increase faster than those of the SNRF (Complex-Only) algorithm (no matter what the team objective is). This observation meets our expectation since robots that use 400 the SARF (Simple-Only and Mixed) algorithms need to construct and evaluate moresimple-targetexchangeswhenthenumberofsimpletargetsincreases. • The response times of the SNRF (Simple-Only and Mixed) algorithms are larger than10secondswhentherobotsimple-targetcapacitiesare5(nomatterwhatthe team objective is). In order to make the SNRF (Simple-Only and Mixed) algo- rithmsrespectthereal-timeconstraint, wereducethediscretizationgranularityG for approximating reaction functions from 30 to 10, and then report the resulting team costs and response times in Table A.117. We make the following observa- tions: – TheteamcostsoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms with discretization granularityG = 10 are slightly larger than those withG = 30 as shown in Table A.86 (no matter what the team objective is). This observation meets our expectation since approximated reaction func- tions are closer to the ideal ones when the discretization granularityG is larger. – TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed) algorithms with discretization granularityG = 10 are smaller than those withG = 30 as shown in Table A.87 (no matter what the team objective is). Thisobservationmeetsourexpectationsincerobotsneedtosolvefewer instancesofthetravelingsalesmanproblemwithtimewindowstoconstruct theapproximatedreactionfunctionswhenthediscretizationgranularityG is smaller. Table A.88 tabulates the team costs of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain with ideal reaction-function calculations. We make thefollowingobservations: 401 • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms with ideal reaction-function calculations are slightly smaller than those with approximated reaction-function calculations shown in Table A.86 (no mat- ter what the team objective is). This observation meets our expectation since ideal reaction functions typically contains more meta-functions than approxi- mated ones. Thus, the auctioneer can find a better allocation (in terms of both coalitions and visit times) and, for a given allocation, a robot can find a better routebyusingidealreactionfunctions. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms with ideal reaction-function calculations are very close to those with approximated reaction-function calculations (no matter what the team objective is). This observation demonstrates that the approximated reaction functions con- structed with the Or-opt heuristic with discretization granularityG = 30 are very closetotheidealones. Table A.89 tabulates the response times of our task-allocation algorithms for dis- joint coalitions in outdoor-like terrain with ideal reaction-function calculations. We makethefollowingobservations: • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithmswithidealreaction-functioncalculationsareslightlylargerthanthosewith approximated reaction-function calculations shown in Table A.87. This obser- vation meets our expectation: i) it takes robots more time to construct the ideal reaction functions since they have to evaluate all possible routes of visiting their allocated targets; and ii) it takes the auctioneer more time to determine the visit times of complex targets since ideal reaction functions typically contain more meta-functionsthantheapproximatedones. 402 Robots Simple Complex Simple-Target SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) Random Targets Targets Capacity Max (Min) Max (Min) Max (Min) Max (Min) Max (Min) MiniSumTeamObjective 10 30 5 ∞ 9.33 (0.07) 9.33 (0.07) 8.80 (0.03) 8.70 (0.03) 5.00 (0.00) MiniMaxTeamObjective 10 30 5 ∞ 5.85 (1.94) 5.85 (1.94) 5.56 (1.89) 5.63 (1.93) 5.00 (0.00) MiniLatTeamObjective 10 30 5 ∞ 5.92 (1.91) 5.92 (1.91) 5.41 (1.99) 5.45 (1.94) 5.00 (0.00) TableA.118: NumberofSimpleTargetsAllocatedtoRobotswithRobotSimple-Target Capacities∞(DisjointCoalitions) • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms with ideal reaction-function calculations are close to those with approx- imated reaction function calculations. This observation demonstrates that the approximated reaction functions constructed with the Or-opt heuristic with dis- cretizationgranularityG =30areveryclosetotheidealones. Table A.90 tabulates the team costs of our task-allocation algorithms with disjoint coalitions in outdoor-like terrain withlooserobotsimple-targetcapacityconstraints. Wemakethefollowingobservations: • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms are all smaller than those of the SARF (Mixed) algorithm for loose robot simple-target capacity constraints (no matter what the team objective is). These results demonstrate that the SNRF (Mixed) algorithm reduces the team costs of the initial allocations produced by the SARF (Mixed) algorithm effectively even whentherobotsimple-targetcapacityconstraintsareloose. • The difference in the team costs of the SARF (Complex-Only, Simple-Only, and Mixed)algorithmswithdifferentrobotsimple-targetcapacitiesissmallwhenthe robotsimple-targetcapacitiesarelargerthan5fortheMiniMaxandMiniLatteam objectives. This observation meets our expectation: Table A.118 tabulates the number of simple targets allocated by our task-allocation algorithms to robots with the robot simple-target capacities∞. It shows that the number of simple 403 targetsallocatedtorobotsislessthan6onaveragefortheMiniMaxandMiniLat team objectives. Thus, the allocations of our task-allocation algorithms remain almost the same when the robot simple-target capacities are larger than 5 for the MiniMaxandMiniLatteamobjectives. Table A.91 tabulates response times of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain withlooserobotsimple-targetcapacityconstraints. Wemakethefollowingobservations: • The response times of the SNRF (Simple-Only and Mixed) algorithms increase faster with the robot simple-target capacities than those of the SNRF (Complex- Only) algorithm (no matter what the team objective is). This observation meets our expectation since robots that use the SNRF (Simple-Only and Mixed) algo- rithmshavetoconstructandevaluatemoresingle-targetexchangeswhentherobot simple-targetcapacityconstraintsareloose. • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms increase faster when the robot simple-target capacities for the MiniSum team objective than for the MiniMax and MiniLat team objectives. This obser- vation meets our expectation: Table A.118 shows that the maximal number of simpletargetsallocatedtorobotsismuchlargerfortheMiniSumteamobjective. Thenumberofvisitordersoftargetsthatrobotsneedtoevaluateforconstructing their reaction functions increases significantly when there are more simple tar- getsallocatedtothem. Thus,robotsneedmoretimeinconstructingtheirreaction functions for the MiniSum team objective when the robot simple-target capacity constraintsareloose. 404 • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithmsremainsmallforlooserobotsimple-targetcapacityconstraints. Forexam- ple, the response times of the SNRF (Mixed) algorithm are smaller than fifteen seconds even when the robot simple-target capacities are∞. This observation demonstrates that the SNRF (Mixed) algorithm re-allocates targets among robots efficientlyevenforlooserobotsimple-targetcapacityconstraints. Table A.92 tabulates the team costs of our task-allocation algorithms for disjoint coalitions in outdoor-like terrain if the initial allocations are produced by the random- izedalgorithm. Wemakethefollowingobservations: • The team costs of the SNRF (Complex-Only) algorithm are smaller than those oftherandomizedalgorithm(nomatterwhattheteamobjectiveis). FortheMin- iSumteamobjective,theSNRF(Complex-Only)algorithmreducestheteamcosts oftherandomizedalgorithmbyonaverage32.51percent. FortheMiniMaxteam objective,theSNRF(Complex-Only)algorithmreducestheteamcostsoftheran- domized algorithm on averageby36.71percent. Forthe MiniLat team objective, the SNRF (Complex-Only) algorithm reduces the team costs of the randomized algorithm on average by 43.55 percent. This observation meets our expectation since the SNRF (Complex-Only) algorithm repeatedly constructs and performs profitable complex-target exchanges between two robots to improve the current allocations. It is guaranteed that the SNRF (Complex-Only) algorithm does not increasetheteamcostsoftheinitialallocationsproducedbytherandomizedalgo- rithm. • TheteamcostsoftheSNRF(Simple-Only)algorithmaresmallerthanthoseofthe SNRF (Complex-Only) algorithm (no matter what the team objective is). For the MiniSum team objective, the SNRF (Simple-Only) algorithm reduces the team 405 costs of SNRF (Complex-Only) algorithm on average by 24.91 percent. For the MiniMax team objective, the SNRF (Simple-Only) algorithm reduces the team costs of the SNRF (Complex-Only) algorithm on average by 18.98 percent. For theMiniLatteamobjective,theSNRF(Simple-Only)algorithmreducestheteam costsoftheSNRF(Complex-Only)algorithmonaverageby36.07percent. Statis- ticalcomparisonssupportthehypothesesthattheteamcostsoftheSNRF(Simple- Only)algorithmarenotlargerthanthoseoftheSNRF(Complex-Only)algorithm whentheinitialallocationsareproducedbytherandomizedalgorithmwithconfi- dence0.9995(whichissignificant). Thisobservationmeetsourexpectationsince theteamcostsoftheSNRF(Simple-Only)algorithmarealsonotlargerthanthose oftheSNRF(Complex-Only)algorithmwhentheinitialallocationsareproduced bytheSARF(Mixed)algorithm,asshowninTableA.80. • TheteamcostsoftheSNRF(Mixed)algorithmaresmallerthanthoseoftheSNRF (Simple-Only)algorithm(nomatterwhattheteamobjectiveis)whenthenumber ofcomplextargetsislargerthanone. FortheMiniSumteamobjective,theSNRF (Mixed) algorithm reduces the team costs of the SNRF (Simple-Only) algorithm by on average 6.29 percent. For the MiniMax team objective, the SNRF (Mixed) algorithm reduces the team costs of the SNRF (Simple-Only) algorithm on aver- age by 3.06 percent. For the MiniLat team objective, the SNRF (Mixed) algo- rithmreducestheteamcostsoftheSNRF(Simple-Only)algorithmonaverageby 5.05 percent. Statistical tests support the hypothesis that the team costs of SNRF (Mixed) algorithm are not larger than those of SNRF (Simple-Only) algorithm whentheinitialallocationsareproducedbytherandomizedalgorithmwithconfi- dence0.9995(whichissignificant). Thisobservationmeetsourexpectationsince the team costs of the SNRF (Mixed) algorithm are also not larger than those of 406 Robots Simple Complex SARF(Complex-Only) SARF(Simple-Only) SARF(Mixed) Targets Targets Profitable Negotiation Profitable Negotiation Profitable Negotiation TaskExchanges Rounds TaskExchanges Rounds TaskExchanges Rounds MiniSumTeamObjective 2 6 1 0.00 1.00 4.19 2.20 4.19 2.20 4 12 2 1.78 1.91 25.23 5.43 27.01 5.73 6 18 3 6.11 3.25 65.65 9.78 71.76 10.60 8 24 4 11.81 4.27 126.03 14.16 137.84 15.15 10 30 5 20.14 5.61 197.41 18.81 217.55 20.33 MiniMaxTeamObjective 2 6 1 0.00 1.00 4.10 2.06 4.10 2.06 4 12 2 1.55 1.80 14.82 4.63 16.37 4.94 6 18 3 3.96 2.44 32.39 6.34 36.35 7.15 8 24 4 6.28 3.12 45.73 7.74 52.01 8.08 10 30 5 8.71 3.46 65.84 8.18 74.55 8.70 MiniLatTeamObjective 2 6 1 0.00 1.00 4.02 2.18 4.02 2.18 4 12 2 2.03 1.94 25.57 5.82 27.60 6.38 6 18 3 6.25 3.07 69.46 10.53 75.71 12.00 8 24 4 11.27 4.16 128.92 15.37 140.19 18.21 10 30 5 20.26 5.45 209.31 21.29 229.57 25.08 Table A.119: Comparison of Numbers of Profitable Task Exchanges and Negotiation RoundsforInitialAllocationsProducedbyRandomizedAlgorithm the SNRF (Simple-Only) algorithm when the initial allocations are produced by theSARF(Mixed)algorithm,asshowninTableA.80. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithmswiththeinitialallocationsproducedbytherandomizedalgorithmarelarger than those of the SNRF algorithms with the initial allocations produced by the SARF(Mixed)algorithmshowninTableA.80(nomatterwhattheteamobjective is). Thisobservationisnotsurprisingsincetheteamcostsoftheinitialallocations produced by the randomized algorithm are several times larger than those of the initialallocationsproducedbytheSARF(Mixed)algorithm. • The team costs of the SNRF (Mixed) algorithm with the initial allocations pro- duced by the randomized algorithm are smaller than those of the SARF (Mixed) algorithmand SKAuctions(no matterwhatthe teamobjectiveis). Thisobserva- tiondemonstratesthattheSNRF(Mixed)algorithmreducedtheteamcostsofthe initialallocationseffectivelynomatterhowtheinitialallocationsareproduced. 407 Table A.93 tabulates the response times of our task-allocation algorithms for dis- joint coalitions in outdoor-like terrain if the initial allocations are produced by theran- domizedalgorithm. Wemakethefollowingobservations: • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms remain small when the initial allocations are produced by the random- ized algorithm (no matter what the team objective is). For example, the response timesoftheSNRF(Mixed)algorithmarelessthanfourseconds. Thisobservation demonstrates that the SNRF (Mixed) algorithm re-allocates targets among robots efficientlynomatterhowtheinitialallocationsareproduced. • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithmswiththeinitialallocationsproducedbytherandomizedalgorithmarelarger than those of the SNRF algorithms with the initial allocations produced by the SARF(Mixed)algorithmshowninTableA.81(nomatterwhattheteamobjective is). This observation meets our expectation since there exist many more prof- itablecomplex-targetandsimple-targetexchangeswhentheinitialallocationsare produced by the randomized algorithm. To improve our understanding of this explanation, we tabulates the numbers of profitable task exchanges and negoti- ation rounds of the SNRF algorithms for the initial allocations produced by the randomized algorithm in Table A.119. Comparing with the results shown in Table A.116 for the initial allocations produced by the SARF (Mixed) algorithm, weobservethatbothnumbersofprofitabletaskexchangesandnegotiationrounds of the SNRF algorithms when the initial allocations produced by the randomized algorithmarelargerthanthatoftheSNRFalgorithmswhentheinitialallocations producedbytherandomizedalgorithm. 408 Table A.94 tabulates the team costs of our task-allocation algorithms for disjoint coalitionsinunknownoutdoor-liketerrain. Wemakethefollowingobservations: • Theteamcostsofourtask-allocationalgorithmsinunknownterrainarealllarger than those in known terrain as shown in Table A.80 (no matter what the team objectiveis). Thisobservationmeets ourexpectationsincerobotsneed torepeat- edly update the blockage status of cells and revise their allocations and routes whenvisitingtargetsinunknownterrain. • TheteamcostsoftheSNRF(Complex-Only,Simple-Only,andMixed)algorithm in unknown terrain are smaller than those of the SARF (Mixed) algorithm (no matter what the team objective is). This observation meets our expectation since the team costs of the SNRF algorithms are also smaller than those of the SNRF (Mixed) algorithm in known terrain. Statistical tests support the hypothesis that the team costs of the SNRF (Mixed) algorithm are not larger than those of the SARF (Mixed) algorithm in unknown terrain with confidence 0.9995 (which is significant). TableA.95tabulatesthenumbersoftherepeatedexecutionsofourtask-allocation algorithms for disjoint coalitions in unknown outdoor-like terrain. We make the fol- lowingobservations: • Thenumbersoftherepeatedexecutionsoftherandomizedalgorithminunknown terrainarelargerthanthoseoftheothertask-allocationalgorithms(nomatterwhat theteamobjectiveis). Thisobservationmeetsourexpectationsincetheteamcosts of the randomized algorithm are much larger than those of other task-allocation algorithms in unknown terrain. Thus, the chance that there are blocked cells in theirrobotroutesbecomeslarger. 409 • The repeated executions of the SNRF (Mixed) algorithm in unknown terrain are smaller than those of the SARF (Mixed) algorithm (no matter what team objec- tive is). This observation meets our expectation since the team costs of the SNRF(Mixed)algorithmsinunknownterrainaresmallerthanthoseoftheSNRF (Mixed)algorithm. Table A.96 tabulates the cumulative response times of our task-allocation algo- rithms for disjoint coalitions in unknown outdoor-like terrain. We make the following observations: • The cumulative response times of the SNRF (Complex-Only, Simple-Only, and Mixed)algorithmsinunknownterrainarelargerthanthoseoftheSARF(Mixed) algorithm (no matter what the team objective is). This observation meets our expectation since the response times of the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms are also larger than those of the SARF (Mixed) algorithm inknownterrain,asshowninTableA.81. • The cumulative response times of the SNRF (Mixed) algorithm in unknown ter- rain are larger than those of the SNRF (Complex-Only and Simple-Only) algo- rithms. This observation meets our expectation since robots that use the SNRF (Mixed)algorithmneedtoconstructandperformbothsimple-targetandcomplex- targetexchangesrepeatedly. TableA.97tabulatestheteamcostsofourtask-allocationalgorithmsforoverlapping coalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: • The team costs of the SNRF (Complex-Only) algorithm are identical to those of theinitialallocationsproducedbytheSARF(Mixed)algorithmandtheteamcosts oftheSNRF(Mixed)algorithmareidenticaltothoseoftheSNRF(Simple-Only) 410 algorithm when the number of robots is two (no matter what the team objective is). This observation meets our expectation since no complex-target exchanges canbeperformedwhenthereareonlytworobots. • TheteamcostsoftheSNRF(Complex-Only)algorithmaresmallerthanthoseof theSARF(Mixed)algorithmwhenthethenumberofrobotsislargerthantwo(no matter what the team objective is). For the MiniSum team objective, the SNRF (Complex-Only) algorithm reduces the team costs of the SARF (Mixed) algo- rithm on average by 1.64 percent. For the MiniMax team objective, the SNRF (Complex-Only) algorithm reduces the team costs of the SARF (Mixed) algo- rithm on average by 9.92 percent. For the MiniLat team objective, the SNRF (Complex-Only) algorithm reduces the team costs of the SARF (Mixed) algo- rithmonaverageby2.89percent. ItisguaranteedthattheSNRF(Complex-Only) algorithm does not increases the team costs of the initial allocations produced by theSARF(Mixed)algorithm. • TheteamcostsoftheSNRF(Simple-Only)algorithmaresmallerthanthoseofthe SNRF (Complex-Only) algorithm (no matter what the team objective is). For the MiniSum team objective, the SNRF (Simple-Only) algorithm reduces the team costsoftheSNRF(Complex-Only)algorithmonaverageby6.85percent. Forthe MiniMax team objective, the SNRF (Simple-Only) algorithm reduces the team costs of the SNRF (Complex-Only) algorithm on average by 14.51 percent. For theMiniLatteamobjective,theSNRF(Simple-Only)algorithmreducestheteam costsoftheSNRF(Complex-Only)algorithmonaverageby5.16percent. Statis- tical tests support the hypothesis that the team costs of the SNRF (Simple-Only) algorithm are not larger than those of the SNRF (Complex-Only) algorithm with confidence 0.9995 (which is significant). This observation meets our expectation 411 sincetheteamcostsoftheSNRF(Simple-Only)algorithmarealsonotlargerthan thoseoftheSNRF(Complex-Only)algorithmfordisjointcoalitions,asshownin TableA.80. • The team costs of the SNRF (Mixed) algorithm are smaller than those of the SNRF(Simple-Only)algorithmwhenthenumberofrobotsislargerthantwo(no matter what the team objective is). For the MiniSum team objective, the SNRF (Mixed) algorithm reduces the team costs of the SNRF (Simple-Only) algorithm on average by 1.26 percent. For the MiniMax team objective, the SNRF (Mixed) algorithm reduces the team costs of the SNRF (Simple-Only) algorithm on aver- age by 2.62 percent. For the MiniLat team objective, the SNRF (Mixed) algo- rithmreducestheteamcostsoftheSNRF(Simple-Only)algorithmonaverageby 2.09 percent. Statistical tests support the hypothesis that the team costs of SNRF (Mixed) algorithm are not larger than those of SNRF (Simple-Only) algorithm with confidence 0.9995 (which is significant). This observation meets our expec- tationsincetheteamcostsoftheSNRF(Mixed)algorithmarealsonotlargerthan those of the SNRF (Simple-Only) algorithm for disjoint coalitions, as shown in TableA.80. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithmsaresmallerthanthoseofSKAuctionsandtherandomizedalgorithm. This observationmeetsourexpectationsincesincetheteamcostsoftheSNRF(Mixed) algorithm are also smaller than those of SK Auctions and the randomized algo- rithm,asshowninTableA.80. Table A.98 tabulates the response times of our task-allocation algorithms for over- lappingcoalitionsinoutdoor-liketerrain. Wemakethefollowingobservations: 412 • Theresponsetimesofourtask-allocationalgorithmsareallsmall(nomatterwhat the team objective is). This observation demonstrates that the the SNRF (Mixed) algorithmre-allocatestargetsamongrobotsefficientlyevenforoverlappingcoali- tions. • TheresponsetimesoftheSNRF(Simple-Only)algorithmarelargerthanthoseof theSNRF(Complex-Only)algorithm(nomatterwhattheteamobjectiveis). This observationmeetsourexpectationsincetheresponsetimesoftheSNRF(Simple- Only)algorithmarealsolargerthanthoseoftheSNRF(Complex-Only)algorithm fordisjointcoalitions,asshowninTableA.81. • The response times of the SNRF (Mixed) algorithm are larger than those of the SNRF (Simple-Only) algorithm (no matter what the team objective is). This observationmeetsourexpectationsincetheresponsetimesoftheSNRF(Mixed) algorithm are also larger than those of the SNRF (Simple-Only) algorithm for disjointcoalitions,asshowninTableA.81. Tables A.99 and A.101 tabulate the team costs of our task-allocation algorithms for overlapping coalitions inempty andindoor-like terrain, respectively. We make the followingobservations: • The team costs of our task-allocation algorithms in empty terrain are all smaller than those in outdoor-like terrain shown in Table A.80, and the team costs of our task-allocation algorithms in indoor-like terrain are all larger than those in outdoor-like terrain shown in Table A.80 (no matter what the team objective is). This observation meets our expectation for the same explanation we made on the resultsshowninTablesA.82andA.84fordisjointcoalitions. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms are smaller than those of the SARF (Mixed) algorithm in both empty and 413 indoor-like terrain (no matter what the team objective is). The results demon- stratesthattheSNRF(Mixed)algorithmreducestheteamcostsoftheinitialallo- cationsproducedbytheSARF(Mixed)algorithmeffectivelyindifferentkindsof terrain. Tables A.100 and A.102 tabulate the response times of our task-allocation algo- rithms for overlapping coalitions in empty andindoor-like terrain, respectively. These results are almost identical to those in outdoor-like terrain shown in Table A.98, and demonstrate that the SNRF (Mixed) algorithm re-allocates targets among robots effi- cientlyindifferentkindsofterrain. Table A.103 tabulates the team costs of our task-allocation algorithms for overlap- ping coalitions in outdoor-like terrain with different robot simple-target capacities. Wemakethefollowingobservations: • The team costs of our task-allocation algorithms all increase when the robot simple-target capacity increases (no matter what the team objective is). This observationmeetsourexpectationforthesameexplanationwemadeontheresults showninTableA.86fordisjointcoalitions. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms are smaller than those of the SARF (Mixed) algorithm for different robot simple-target capacities (no matter what the team objective is). The results demonstrate that the SNRF (Mixed) algorithm reduces the team costs of the ini- tialallocationsproducedbytheSARF(Mixed)algorithmeffectivelyfordifferent robotsimple-targetcapacities. TableA.104tabulatestheresponsetimesofourtask-allocationalgorithmsforover- lappingcoalitionsinoutdoor-liketerrainwithdifferentrobotsimple-targetcapacities. Wemakethefollowingobservations: 414 • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms increase when the robot simple-target capacity increases (no matter what theteamobjectiveis). Thisobservationmeetsourexpectationsinceittakesrobots moretimetoconstructtheirgeneralizedreactionfunctionswhentheyneedtovisit moresimpletargets. • Theresponsetimesofourtask-allocationalgorithmsremainsmallwhentherobot simple-target capacity increases (no matter what the team objective is). These results demonstrate that the SNRF (Mixed) algorithm re-allocates targets among robotsefficientlyfordifferentrobotsimple-targetcapacities. Table A.105 tabulates the team costs of our task-allocation algorithms for overlap- ping coalitions in outdoor-like terrain with ideal reaction-function calculations. We makethefollowingobservations: • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms with ideal reaction-function calculations are slightly smaller than those with approximated reaction-function calculations shown in TableA.103 (no mat- ter what the team objective is). This observation meets our expectation for the same explanation we made on the results shown in Table A.86 for disjoint coali- tions. • The team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algo- rithms with ideal reaction-function calculations are very close to those with approximated reaction-function calculations (no matter what the team objective is). This observation demonstrates that the approximated reaction functions con- structed by the Or-opt heuristic described in Subsection 5.4.2 are very similar to theidealones. 415 Robots Simple Complex Simple-Target SARF(Mixed) SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) Random Targets Targets Capacity Max (Min) Max (Min) Max (Min) Max (Min) Max (Min) MiniSumTeamObjective 10 30 3 ∞ 9.17 (0.36) 9.15 (0.38) 8.91 (0.34) 8.90 (0.34) 5.00 (0.00) MiniMaxTeamObjective 10 30 3 ∞ 5.07 (1.51) 5.08 (1.50) 4.87 (1.57) 4.84 (1.58) 5.00 (0.00) MiniLatTeamObjective 10 30 3 ∞ 5.00 (1.58) 4.97 (1.58) 4.85 (1.71) 4.83 (1.74) 5.00 (0.00) TableA.120: NumberofSimpleTargetsAllocatedtoRobotswithRobotSimple-Target Capacities∞(OverlappingCoalitions) TableA.106tabulatestheresponsetimesofourtask-allocationalgorithmsforover- lappingcoalitionsinoutdoor-liketerrainwithidealreaction-functioncalculations. We makethefollowingobservations: • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithmswithidealreaction-functioncalculationsarelargerthanthosewithapprox- imated reaction-function calculations shown in Table A.104. This observation meets our expectation for the same explanation we made on the results shown in TableA.89fordisjointcoalitions. • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms with ideal reaction-function calculations are close to those with approx- imated reaction function calculations shown in Table A.104. This observation canbeexplainedbytheapproximatedreactionfunctionsbeingverysimilartothe idealones. Table A.107 tabulates the team costs of our task-allocation algorithms for overlap- ping coalitions in outdoor-like terrain with loose robot simple-target capacity con- straints. Wemakethefollowingobservations: • TheteamcostsoftheSNRF(Complex-Only,Simple-Only,andMixed)algorithm are smaller than those of the SARF (Mixed) algorithm for loose robot simple- target capacity constraints (no matter what the team objective is). The results 416 Robots Simple Complex Simple-Target SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed)) Targets Targets Capacities TeamCost ResponseTime TeamCost ResponseTime TeamCost ResponseTime MiniSumTeamObjective 10 50 3 ∞ 319.89 0.12 310.76 5.35 310.09 5.58 MiniMaxTeamObjective 10 50 3 ∞ 55.35 0.01 50.61 0.04 49.90 0.05 MiniLatTeamObjective 10 50 3 ∞ 686.52 0.01 667.64 0.06 664.90 0.09 TableA.121: ComparisonofDifferentDiscretizationGranularities(OverlappingCoali- tions) demonstratethattheSNRF(Mixed)algorithmreducestheteamcostsoftheinitial allocations produced by the SARF (Mixed) algorithm effectively even when the robotsimple-targetcapacityconstraintsareloose. • The difference in the team costs of the SNRF (Complex-Only, Simple-Only, and Mixed) algorithms with different number of robot simple-target capacity becomes small when the robot simple-target capacities are larger than 5 for the MiniMax and MiniLat team objectives. This observation meets our expectation: Table A.120 tabulates the number of simple targets allocated allocated by our task-allocation algorithms to robots with the robot simple-target capacities∞. It showsthatthenumberofsimpletargetsallocatedtorobotsisonaveragelessthan 6 for the MiniMax and MiniLat team objectives. Thus, the allocations of our task-allocation algorithms remain almost the same when the robot simple-target capacitiesarelargerthan5fortheMiniMaxandMiniLatteamobjectives. TableA.108tabulatestheresponsetimesofourtask-allocationalgorithmsforover- lappingcoalitionsinoutdoor-liketerrainwithlooserobotsimple-targetcapacitycon- straints. Wemakethefollowingobservations: • TheresponsetimesoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms increase faster with the robot simple-target capacities for the MiniSum 417 teamobjectivethanfortheMiniMaxandMiniLatteamobjectives. Thisobserva- tionmeetsourexpectationforthesameexplanationwemadeontheresultsshown inTableA.91fordisjointcoalitions. • The response times of SNRF (Simple-Only and Mixed) algorithms for robot simple-target capacities∞ are larger than 20 seconds for the MiniSum team objective. In order to make the SNRF (Simple-Only and Mixed) algorithms respect the real-time constraints, we reduce the discretization granularityG for approximating generalized reaction functions from 30 to 10 and then report the resulting team costs and response times in Table A.121. We make the following observations: – TheteamcostsoftheSNRF(Complex-Only,Simple-Only,andMixed)algo- rithms with discretization granularityG = 10 are slightly larger than those withG = 30 shown in Table A.107 (no matter what the team objective is). This observation meets our expectation since the approximated reac- tionfunctionsareclosertotheidealoneswhenthediscretizationgranularity islarger. – The response times of the SNRF (Simple-Only and Mixed) algorithms with discretization granularityG = 10 are smaller than those withG = 30 shown in Table A.108 (no matter what the team objective is). This observation meetsourexpectationsincerobotsneedtosolvefewerinstancesoftraveling salesman problems with time windows to construct approximated reaction functionswhenthediscretizationgranularityissmaller. Table A.109 tabulates the team costs of our task-allocation algorithms for overlap- ping coalitions in outdoor-like terrain with different numbers of complex targets. We makethefollowingobservations: 418 Robots Simple Complex SNRF(Complex-Only) SNRF(Simple-Only) SNRF(Mixed) Targets Targets Constructing Determining Total Constructing Determining Total Constructing Determining Total ReactionFunctions VisitTimes ReactionFunctions VisitTimes ReactionFunctions VisitTimes MiniSumTeamObjective 10 30 7 2.75 40.98 43.73 3.07 28.84 31.91 10.14 94.61 104.75 MiniMaxTeamObjective 10 30 7 0.43 4.35 4.78 2.09 1.31 3.40 2.83 4.93 7.76 MiniLatTeamObjective 10 30 7 5.47 146.50 151.97 5.10 72.42 77.52 16.56 351.02 367.58 Table A.122: Response Times of SNRF Algorithms with 7 Complex Targets (Overlap- pingCoalitions) • Theteamcostsofourtask-allocationalgorithmsallincreasewhenthenumberof complextargetsincreases. Thisobservationmeetsourexpectationsincetherobot costusuallyincreaseswhenarobothastovisitalargenumberofcomplextargets. • TheteamcostsoftheSNRF(Complex-Only,Simple-OnlyandMixed)algorithms are smaller than those of the SARF (Mixed) algorithm for different numbers of complex targets (no matter what the team objective is). The results demonstrate that the SNRF (Mixed) algorithm reduces the team costs of the initial allocations produced by the SARF (Mixed) algorithm effectively even for different numbers ofcomplextargets. Table A.110 tabulates the response times of our