The Traveling Salesman Problem: State of the Art

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TheTravelingSalesmanProblem:
StateoftheArt
ThomasSt
¨
utzle
stuetzle@informatik.tu-darmstadt.de
http://www.intellektik.informatik.tu-darmstadt.de/˜tom.
DarmstadtUniversityofTechnology
DepartmentofComputerScience
IntellecticsGroup
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.1
Outline
TSP
benchmarks
completealgorithms
constructionheuristics
localsearch
SLSmethods,ILS
concludingremarks
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.2
TravelingSalesmanProblem
given
:fullyconnected,
weightedGraph


 





goal
:findshortest
Hamiltoniancycle
hardness
:

-hard
interest
:standard
benchmarkproblemfor
algorithmicideas
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.3
WhyTSP?
TheTSPisprobablythemostwidelystudied
combinatorialoptimizationproblem
conceptuallysimpleproblem
hardtosolve(

-hard)
didactic,designandanalysisofalgorithmsnotobscuredby
technicalities
significantamountofresearch
arisesinavarietyofapplications
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.4
Benchmarkinstances
Generalconsiderations
mainlymetricTSPs,oftenEuclidean
typicallyonlyintegerdistances
Instances
TSPLIB
morethan100instancesupto85.900cities
someinstancesfrompracticalapplications
instancesfromVLSIdesign
randomEuclideaninstances(uniformandclustered)
someavailablefrom8thDIMACSchallenge
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.5
Benchmarkinstances
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.6
Benchmarkinstances
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.7
Benchmarkinstances
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.8
Benchmarkinstances
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.9
Lowerbounds
lowerboundsarenecessarytoratequalityguaranteesof
tours
areneededwithincompletealgorithms
bestlowerbound:Held-Karpbounds
experimentally,showntobeverytight
withinlessthanonepercentofoptimumforrandom
Euclidean
uptotwopercentforTSPLIBinstances
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.10
Completealgorithms
branch&boundalgorithms
branch&cutalgorithms(state-of-the-art)
useLP-relaxationforlowerboundingschemes
effectiveheuristicsforupperbounds
branchifcutscannotbefoundeasily
state-of-the-art
largestinstancesolvedhas15.112cities!
ofteninstanceswithfewthousandsofcitiescanbe
regularilysolvedwithinminutes/hours
Concordecodepublicallyavailable
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.11
Completealgorithms
SolutiontimeswithConcorde
InstanceNo.nodesCPUtime(secs)
att5327109.52
rat783137.88
pcb117319468.27
fl157776705.04
d210516911179253.91
pr23921116.86
rl5934205588936.85
usa135099539ca.4years
d15112164569ca.22years
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.12
Constructionheuristics
growtoursbyiterativelyaddingcitiestopartialtours
manyvariantsavailable
nearestneighborheuristics
(ca.22%fromopt)
insertionheuristics
(ca.14%fromopt;farthestinsertion)
greedyheuristics
(ca.14%fromopt)
savingsheuristic
(ca.12%fromopt)
Christofidesheuristic
(ca.10%fromopt)
goodcompromisebetweensolutionqualityandcomputation
timebygreedyorsavingsheuristic
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.13
Localsearchalgorithms

-exchangeheuristics
2-opt
2.5-opt
3-opt
complexneighborhoods
Lin-Kernighan
Helsgaun’sLin-Kernighanvariant
Dynasearch
ejectionchainsapproach
allexploitTSPspecificandgeneralimplementationand
speed-uptechniques
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.14
Implementation
typicallyallalgorithmsusefirst-improvement
neighborhoodpruning
fixedradiusnearestneighborsearch
neighborlists
restrictexchangestomostinterestingcandidates
don’tlookbits
focuslocalsearchto"interesting"part
sophisticateddatastructures
extremelylargeinstancestackable
(largesthad25millioncities!)
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.15
Exampleresults:TSP
timingsfor1000localsearcheswith2-optand3-optvariantsfromrandominitial
solutionsonaPentiumIII500MHzCPU.std:nospeed-uptechniques;fr+cl:fixedradius
andunboundedcandidatelists,dlb:don’tlookbits
2-opt-std
2-opt-fr+cl
2-opt-fr+cl+dlb
3-opt-fr+cl+dlb
instance

avg
 

avg
 

avg
 

avg
 
kroA100
8.91.6
6.40.5
6.60.4
2.44.3
d198
5.76.4
4.21.2
4.30.8
1.430.1
lin318
10.622.1
7.52.1
7.91.5
3.465.5
pcb442
12.755.7
7.12.9
7.62.2
3.863.4
rat783
13.0239.7
7.57.5
8.05.8
4.2213.8
pr1002
12.8419.5
8.413.2
9.29.7
4.6357.6
pcb1173
14.5603.1
8.516.7
9.312.4
5.2372.3
d1291
16.8770.3
10.116.9
11.112.4
5.5377.6
fl1577
13.61251.1
7.925.8
9.019.2
4.0506.8
pr2392
15.02962.8
8.865.5
10.149.1
5.3878.1
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.16
Lin-Kernighanheuristic
complexmovesarebuildasbeingaconcatenationofa
numberofsimplemoves
thenumberofsimplemovescomposingacomplexoneis
variableanddeterminedbasedongaincriteria
thesimplemovesneednotbeindependentofeachother
terminationisguaranteedthroughadditionalconditionson
thesimplemoves
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.17
Exampleresults:3-optandLK
takenfromDIMACSChallengeresults,normalizedtimesonaCompaqDS20500MhzAlpha
EV6
3-opt-JM
LK-JM
LK-ABCC
LK-H
instance

avg
 

avg
 

avg
 

avg
 
pcb1173
3.090.12
0.870.25
2.140.10
0.186.69
fl1577
5.650.20
0.938.72
9.010.15
5.5614.86
pr2392
3.340.27
1.850.77
3.300.18
0.3434.87
rl5915
2.440.83
1.223.29
3.320.46
0.39242.99
d15112
2.301.99
1.134.60
1.822.38
0.111515.99
pla85900
3.806.87
1.2146.20
1.218.84
0.8548173.84
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.18
Lin-Kernighanheuristic
goodLin-Kernighanimplementationsarethebest
performinglocalsearchesforTSP
manyvariantsofthealgorithmareavailable(seealsorecent
DIMACSchallenge)
anefficientimplementationrequiressophisticateddata
structures(severalarticlesavailableonthissubject)
implementationisquitetimeconsuming,but:
atleastthreeverygoodimplementationsarepublically
available(Concorde,Helsgaun,Neto)
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.19
HybridSLSmethods
HybridSLSmethodsarerequiredwhenveryhighsolutionis
desired
available(goodworking)approaches
iteratedlocalsearch
approachesusingILSasasubroutine
tourmerging
multi-levelalgorithms
memeticalgorithms
antcolonyoptimization
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.20
ILSExample—TSP
basicILSalgorithmforTSP
GenerateInitialSolution:greedyheuristic
LocalSearch:2-opt,3-opt,LK,(whateveravailable)
Perturbation:double-bridgemove(a4-optmove)
AcceptanceCriterion:accept



onlyif












ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.21
ILSforTSPs
iterateddescent
Baum,1986
firstapproach,relativelypoorresults
largestepMarkovchains
Martin,Otto,Felten,1991,1992,1996
firsteffectiveILSalgorithmforTSP
introduceddouble-bridgemoveforILS
simulatedannealingtypeacceptancecriterion
iteratedLin-Kernighan
Johnson,1990,1997
efficientILSimplementationbasedonpreprintsofMOF91
efficientLKimplementation
acceptsonlyshortertours
slightlydifferentperturbationfromMOF
dataperturbation
Codenottiet.al,1996
complexperturbationbasedonchangingproblemdata
goodLKimplementation
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.22
ILSforTSPs
improvedLSMC
Hong,Kahng,Moon,1997
studyofdifferentperturbationsizes,acceptancecriteriawith2-opt,3-opt,LKlocal
search
CLOimplementationinConcorde
Applegate,Bixby,Chvatal,Cook,Rohe,199?-today
veryfastLKimplementation,publiclyavailable,appliedtoextremelylargeinstances
(25millioncities!)
variousperturbationsavailable
ILSwithfitness-distancebaseddiversification
Stützle,Hoos1999
diversificationmechansiminILSforlongruntimes
verygoodperformancewithonly3-optlocalsearch
ILSwithgenetictransformation
Katayama,Narisha,1999
perturbationguidedbyasecondsolution
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.23
ILSforTSPs
IteratedHelsgaun
extremelyeffectiveLKimplementationbasedon5-exchangemoves
constructivemechanismforgeneratingnewstartingtours(nodouble-bridgemove)
newwayofconstructingverysmallcandidatelists
foundmanynewbesttoursforlargeinstances
IteratedLKvariantbyNguyenetal.
effectiveLKimplementationbasedonHelsgaun’sideas
muchfasterthanIteratedHelsgaun
newwayofconstructingverysmallcandidatelists
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.24
MemeticalgorithmsforTSPs
Justtomentionafew..
Gorges-Schleuter,Mühlenbein(1989)
Braun(1992?)
Nagata,Kobayashi(edge-assemblycrossover,1997)
Merz,Freisleben(DPX-crossover,greedycrossover,
1996–2002)
Möbiusetal.(Iterativepartialtranscription,1999)
Houdayer,Martin(recursiveMA,1999)
itisnotclearwhetherthesealgorithmsareabletobeatgoodILS
algorithmswhenveryeffectivelocalsearchesareused
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.25
Exampleresults:ILSandextensions
takenfromDIMACSChallengeresults,normalizedtimesonaCompaqDS20500MhzAlpha
EV6
ILS-JM
ILS-ABCC
ILS-H
TourMerging
instance

avg
 

avg
 

avg
 

avg


pcb1173
0.147.09
0.281.45
0.1823.6
0.030.8
fl1577
0.69311.76
7.056.0
0.06440.5
0.02161.7
pr2392
0.1619.2
0.523.5
0.0150.9
0.092.5
rl5915
0.48129.7
0.7713.9
0.041118.2
0.02704.5
d15112
0.24332.1
0.2074.6
0.0226322.9
——
pla85900
0.276127.8
0.29500.0
——-
——
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.26
State-of-the-art
GeneralresultsforTSPalgorithms
completealgorithmscansolvesurprisinglylargeinstances
constructionheuristicsanditerativeimprovementcanbe
appliedtoverylargeinstances(



cities)with
considerablesuccess
bestperformanceresultsw.r.t.solutionqualityobtained
withiteratedlocalsearch,geneticalgorithms,ormore
TSP-specificapproaches
bestperforminghybridalgorithmsuse3-opt
orLin-Kernighanlocalsearch
8thDIMACSImplementationChallengeontheTSPgives
overviewofstate–of–the–artresults
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.27
Conclusions
TheTSP...
hasbeenasourceofinspirationfornewalgorithmicideas
isastandardtest-bedforcompleteandincomplete
algorithms
RecentcontributionstoTSPsolving:
pushingthefrontieroftractableinstancesize
findingcandidatesolutionsofveryhighquality
betterunderstandingofalgorithmbehaviour
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.28
References
HolgerHoosandThomasStützle.StochasticLocalSearch,Chapter8(TSP),Morgan
KaufmannPublishers,2004.
D.S.JohnsonandL.A.McGeoch.Thetravellingsalesmanproblem:Acasestudyinlocal
optimization.InE.H.L.AartsandJ.K.Lenstra,editors,LocalSearchinCombinatorial
Optimization,pages215–310.JohnWiley&Sons,Chichester,UK,1997.
D.S.JohnsonandL.A.McGeoch.ExperimentalanalysisofheuristicsfortheSTSP.In
G.GutinandA.Punnen,editors,TheTravelingSalesmanProblemanditsVariations,pages
369–443.KluwerAcademicPublishers,2002.
K.Helsgaun.Aneffectiveimplementationofthelin-kernighantravelingsalesmanheuristic.
EuropeanJournalofOperationalResearch,126:106–130,2000.
D.Applegate,R.Bixby,V.Chvátal,andW.Cook.FindingtoursintheTSP.TechnicalReport
99885,ForschungsinstitutfürDiskreteMathematik,UniversityofBonn,Germany,1999.
P.MerzandB.Freisleben.Memeticalgorithmsforthetravelingsalesmanproblem.Complex
Systems,4:297–345,2001.
ThomasSt¨utzle,TheTravelingSalesmanProblem:StateoftheArtTUD—SAPAGWorkshoponVehicleRouting,July10,2003–p.29