The Traveling Salesman Problem: a Computational Study

606 Pages·2011·31.433 MB·English

by  Chvatal, Vasek;Applegate, David L.;Bixby, Robert E.;Cook, William J

The Traveling Salesman Problem
K
O
A Computational Study
O
B
-
David L. Applegate
E
Robert E. Bixby
S
S Vasˇek Chv a´tal
E
William J. Cook
R
P

Y
T
I
S
R
E
V
I
N
U

N
O
T
E
C
N
I
R
P

A
tspbook September11,2006
The Traveling Salesman Problem
tspbook September11,2006
PrincetonSeriesinAppliedMathematics
Editors
Ingrid Daubechies (Princeton University); Weinan E (Princeton University); Jan
KarelLenstra(EindhovenUniversity);EndreSli(UniversityofOxford)
ThePrincetonSeriesinAppliedMathematicspublisheshighqualityadvancedtexts
and monographs in all areas of applied mathematics. Books include those of a
theoretical and general nature as well as those dealing with the mathematics of
specificapplicationsareasandreal-worldsituations.
TitlesintheSeries
ChaoticTransitionsinDeterministicandStochasticDynamicalSystems: Applica-
tions of Melnikov Processes in Engineering, Physics, and Neuroscience by Emil
Simiu
SelfsimilarProcessesbyPaulEmbrechtsandMakotoMaejima
Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms by
JimingPeng,CornelisRoos,andTamsTerlaky
AnalyticTheoryofGlobalBifurcation: AnIntroductionbyBorisBuffoniandJohn
Toland
EntropybyAndreasGreven,GerhardKeller,andGeraldWarnecke
AuxiliarySignalDesignforFailureDetectionbyStephenL.CampbellandRamine
Nikoukhah
Thermodynamics: ADynamicalSystemsApproachbyWassimM.Haddad,Vijay-
SekharChellaboina,andSergeyG.Nersesov
Optimization:InsightsandApplicationsbyJanBrinkhuisandVladimirTikhomirov
MaxPlusatWorkbyBerndHeidergott,GeertJanOlsder,andJacobvanderWoude
GenomicSignalProcessingbyEdDoughertyandIlyaShmulevich
TheTravelingSalesmanProblembyDavidL.Applegate, RobertE.Bixby, Vasˇek
Chva´tal,andWilliamJ.Cook
tspbook September11,2006
The Traveling Salesman
Problem
AComputationalStudy
David L. Applegate
Robert E. Bixby
Vasˇek Chva´tal
William J. Cook
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
tspbook September11,2006
Copyright(cid:2)c2006byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom: PrincetonUniversityPress,3MarketPlace,
Woodstock,OxfordshireOX201SY
AllRightsReserved
LibraryofCongressControlNumber: 2006931528
ISBN-13: 978-0-691-12993
ISBN-10: 0-691-12993-2
Thepublisherwouldliketoacknowledgetheauthorsofthisvolumeforproviding
thecamera-readycopyfromwhichthisbookwasprinted.
BritishLibraryCataloging-in-PublicationDataisavailable
Printedonacid-freepaper.
pup.princeton.edu
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
tspbook September11,2006
Bashonregardless.
J.P.Donleavy,TheDestiniesofDarcyDancer,Gentleman
tspbook September11,2006
tspbook September11,2006
Contents
Preface xi
Chapter1. TheProblem 1
1.1 TravelingSalesman 1
1.2 OtherTravelers 5
1.3 Geometry 15
1.4 HumanSolutionoftheTSP 31
1.5 EngineofDiscovery 40
1.6 IstheTSPHard? 44
1.7 MilestonesinTSPComputation 50
1.8 OutlineoftheBook 56
Chapter2. Applications 59
2.1 Logistics 59
2.2 GenomeSequencing 63
2.3 ScanChains 67
2.4 DrillingProblems 69
2.5 AimingTelescopesandX-Rays 75
2.6 DataClustering 77
2.7 VariousApplications 78
Chapter3. Dantzig,Fulkerson,andJohnson 81
3.1 The49-CityProblem 81
3.2 TheCutting-PlaneMethod 89
3.3 PrimalApproach 91
Chapter4. HistoryofTSPComputation 93
4.1 Branch-and-BoundMethod 94
4.2 DynamicProgramming 101
4.3 GomoryCuts 102
4.4 TheLin-KernighanHeuristic 103
4.5 TSPCuts 106
4.6 Branch-and-CutMethod 117
4.7 Notes 125
Chapter5. LPBoundsandCuttingPlanes 129
5.1 GraphsandVectors 129
5.2 LinearProgramming 131
tspbook September11,2006
viii CONTENTS
5.3 OutlineoftheCutting-PlaneMethod 137
5.4 ValidLPBounds 139
5.5 Facet-InducingInequalities 142
5.6 TheTemplateParadigmforFindingCuts 145
5.7 Branch-and-CutMethod 148
5.8 HypergraphInequalities 151
5.9 SafeShrinking 153
5.10 AlternativeCallstoSeparationRoutines 156
Chapter6. SubtourCutsandPQ-Trees 159
6.1 ParametricConnectivity 159
6.2 ShrinkingHeuristic 164
6.3 SubtourCutsfromTourIntervals 164
6.4 Padberg-RinaldiExactSeparationProcedure 170
6.5 StoringTightSetsinPQ-trees 173
Chapter7. CutsfromBlossomsandBlocks 185
7.1 FastBlossoms 185
7.2 BlocksofG∗ 187
1/2
7.3 ExactSeparationofBlossoms 191
7.4 Shrinking 194
Chapter8. CombsfromConsecutiveOnes 199
8.1 ImplementationofPhase2 202
8.2 ProofoftheConsecutiveOnesTheorem 210
Chapter9. CombsfromDominoes 221
9.1 PullingTeethfromPQ-trees 223
9.2 NonrepresentableSolutionsalsoYieldCuts 229
9.3 Domino-ParityInequalities 231
Chapter10. CutMetamorphoses 241
10.1 Tighten 243
10.2 Teething 248
10.3 Naddef-ThienelSeparationAlgorithms 256
10.4 Gluing 261
Chapter11. LocalCuts 271
11.1 AnOverview 271
11.2 MakingChoicesofV andσ 272
11.3 RevisionistPolicies 274
11.4 Doesφ(x∗)LieOutsidetheConvexHullofT? 275
11.5 Separatingφ(x∗)fromT:TheThreePhases 289
11.6 PHASE1:FromT∗toT(cid:3)(cid:3) 291
11.7 PHASE2:FromT(cid:3)(cid:3)toT(cid:3) 315
11.8 ImplementingORACLE 326
11.9 PHASE3:FromT(cid:3)toT 329
11.10Generalizations 339
tspbook September11,2006
CONTENTS ix
Chapter12. ManagingtheLinearProgrammingProblems 345
12.1 TheCoreLP 345
12.2 CutStorage 354
12.3 EdgePricing 362
12.4 TheMechanics 367
Chapter13. TheLinearProgrammingSolver 373
13.1 History 373
13.2 ThePrimalSimplexAlgorithm 378
13.3 TheDualSimplexAlgorithm 384
13.4 ComputationalResults:TheLPTestSets 390
13.5 Pricing 404
Chapter14. Branching 411
14.1 PreviousWork 411
14.2 ImplementingBranchandCut 413
14.3 StrongBranching 415
14.4 TentativeBranching 417
Chapter15. TourFinding 425
15.1 Lin-Kernighan 425
15.2 FlipperRoutines 436
15.3 EngineeringLin-Kernighan 449
15.4 ChainedLin-KernighanonTSPLIBInstances 458
15.5 Helsgaun’sLKHAlgorithm 466
15.6 TourMerging 469
Chapter16. Computation 489
16.1 TheConcordeCode 489
16.2 RandomEuclideanInstances 493
16.3 TheTSPLIB 500
16.4 VeryLargeInstances 506
16.5 TheWorldTSP 524
Chapter17. TheRoadGoesOn 531
17.1 CuttingPlanes 531
17.2 TourHeuristics 534
17.3 DecompositionMethods 539
Bibliography 541
Index 583