Dueckrewrite规则

Acompletealgorithmtosolvethegraph-coloringproblemHubertoAyaneguiandAlbertoChavez-AragonFacultaddeCienciasBasicas,IngenieriayTecnologia,UniversidadAutonomadeTlaxcala,CalzadadeApizaquitos/n,Apizaco,Tlaxcala,Mexico{hayanegui,albertochz}@gmail.
comAbstract.
TheGraphk-ColorabilityProblem(GCP)isawellknownNP-hardproblemwhichconsistinfindingthekminimumnumberofcolorstopainttheverticesofagraphinsuchawaythatanytwoverticesjoinedbyanedgehavealwaysdifferentcolors.
Manyyearsago,SimulatedAnnealing(SA)wasusedforgraphcoloringtaskobtaininggoodresults;howeverSAisnotacompletealgorithmanditnotalwaysgetstheoptimalsolution.
InthispaperGCPistransformedintotheSatisfiabilityProblemandthenitissolvedusingaalgorithmthatusestheThresholdAcceptingalgorithm(avariantofSA)andtheDavis&Putnamalgorithm.
Thenewalgorithmisacompleteoneandsoitgetsbetterqualitythattheclassicalsimulatedannealingalgorithm.
Keywords:graphcoloring,simulatedannealing,thresholdaccepting,davis&putnam.
1IntroductionLetG=(V,E)beagraphwhereVisasetofverticesandEisasetofedges.
Ak-coloringofGisapartitionofVintoksets{V1,…,Vk},suchthatnotwoverticesinthesamesetareadjacent,i.
e.
,ifv,wbelongtoVi,1ik,then(v,w)notbelongtoE.
Thesets{V1,…,Vk}arereferredtoascolors.
Thechromaticnumber,x(G),isdefinedastheminimumkforwhichGisk-colorable.
TheGraphk-ColorabilityProblem(GCP)canbestatedasfollows.
GivenagraphG,findx(G)andthecorrespondingcoloring.
GCPisaNP-hardproblem[1].
GCPisveryimportantbecauseithasmanyapplications;someofthemareplanningandschedulingproblems[2][3],timetabling[4],mapcoloring[5]andmanyothers.
SinceGCPisaNP-hardproblem,untilnowtherearenotknowndeterministicmethodsthatcansolveditinapolynomialtime[1].
Sonon-deterministicalgorithmshavebeenbuilttosolveit;oneofthemisSimulatedAnnealing(SA)[6]thathasbeenusedonGCPwithgoodresults[7][8].
However,SAisnotacompletealgorithmanditnotalwaysgetstheoptimalsolution.
TheapproachusedinthispaperistotransformGCPintoaSatisfiabilityProblem(orSATproblem)[12]andthenusethealgorithmproposedinthispaper.
WeproposetouseiterativelytheThresholdAccepting(TA)algorithm(avariantofSimulatedAnnealing)[9]andthenaDavisandPutnamalgorithm[10].
2SimulatedannealingandthresholdacceptingSimulatedannealing(SA)[6]isastochasticcomputationaltechniquederivedfromstatisticalmechanicstofindnearglobal-minimum-costsolutionstolargeoptimizationproblems.
Inmanyinstances,findingtheglobalminimumvalueofanobjectivefunctionwithmanydegreesoffreedomsubjecttoconflictingconstraintsisanNP-completeproblem,sincetheobjectivefunctionwilltendtohavemanylocalminimums.
Aprocedureforsolvingoptimizationproblemsofthistypeshouldsamplethesearchspaceinsuchawaythatithasahighprobabilityoffindingtheoptimaloranear-optimalsolutioninareasonabletime.
Overthepastdecade,SAhasproventobeapowerfultechniquethatmeetsthesecriteriaforawiderangeofproblems.
SAexploitsananalogybetweenthewayametalcoolandfreezesintoaminimumenergycrystallinestructure(theannealingprocess)andthesearchforaminimuminamoregeneralsystem.
SAmakesarandomsearchwhichnotonlyacceptschangesthatincreaseitscostfunctionf,butalsosomethatdecreaseit.
Forthisreason,SAusesacontrolparameterc,whichbyanalogywiththeoriginalapplicationisknownasthe"SystemTemperature",cstartsouthighandgraduallydecreases.
AdeterioratingrandommovefromsolutionSitoSjisacceptedwithaprobabilityexp-(f(Sj)-f(Si))/c.
Ifthismoveisnotdeteriorating(thenewsolutionSjisbetterthanthepreviousoneSi)thenitisacceptedandanewrandommoveisproposedagain.
Whenthetemperatureishigh,abadmovecanbeaccepted.
Asctendstozero,SAbecomesmoredemandingthroughacceptjustbettermoves.
Thealgorithmforminimizationisshownbelow:ProcedureSIMULATEDANNEALINGBeginINITIALIZE(Si=initial_solution,c=initial_temperature)k=0RepeatRepeatSj=PERTURBATION(Si)IfCOST(Sj)random[0,1)ThenSi=SjUntilstochasticequilibriumk=k+1c=COOLING(c)UntilthermalequilibriumEndTheINITIALIZEfunctionstartstheinitialsolutionSiandtheinitialtemperaturec.
ThePERTURBATIONfunctionmakesarandomperturbationfromSitogenerateaneighborhoodsolutionSj.
TheCOSTfunctiongetsthecostfromasolution.
TheINC_COSTfunctiongetsthedifferenceincostbetweenSjandSi.
Finally,theCOOLINGfunctiondecreasestheactualtemperatureparameterc.
AvariantofSimulatedAnnealing(SA)istheThresholdAcceptingmethod(TA).
ItwasdesignedbyDueck&Scheuer[9]inordertogetamoreefficientalgorithmthanSimulatedAnnealing.
Theprincipaldifference,betweenSAandTA,isthemechanismofacceptingthesolutionrandomlychosenfromthesetofneighborsofthecurrentsolution.
WhileSAusesaprobabilisticmodel(seeequation(1)),TAusesastaticmodel:ifthedifferencebetweenthecostvaluesofthechosensolutionSjandthecurrentoneSiissmallerthanathresholdT(ortemperature),TAacceptsmovingtothechosensolution.
Otherwiseitstaysatthecurrentsolution.
Again,thethresholdparameterTisapositivecontrolparameterwhichdecreaseswithincreasingnumberofiterationsandconvergestovaluenearto0.
Henceforth,ineveryiterationsomesolutiondeteriorationareallowed;thisdeteriorationdependsonthecurrentthresholdT(seeequation(2));inthiswayonlyimprovingsolutionswithalmostnonedeteriorationsolutionareacceptedattheendoftheprocess.
p(S1,S2)=exp(min{f(S1)-f(S2),0}/c)(1)COST(Sj)ThiscouldbebecauseTAdoesnotcomputetheprobabilisticfunction(1)anddoesnotexpendalotoftimemakingrandomdecisions.
TheThresholdAcceptingalgorithmforminimizationisthefollowing:ProcedureTHRESHOLDACCEPTINGBeginINITIALIZE(Si=initial_solution,T=initial_thresholdortemperature)k=0RepeatRepeatSj=PERTURBATION(Si)E=COST(Sj)–COST(Si)IfESALAusesSimulated-annealingapproachwithtwomainloops:internalloopnamedMetropolisCycleandexternalloopcalledTemperatureCycle.
Numberofiterationsininternalandexternalloopusuallyaretunedexperimentally[6],[9].
However,recentlyananalyticalmethodusingaMarkovmodelwasproposedtotuneTAsolvingSATproblems.
ExternalloopexecutedfromainitialtemperatureTiuntilafinaltemperatureTfandtheinternalloopbuildsaMarkovchainoflengthLkwhichdependsonthetemperaturevalueTk(krepresentsthesequenceindexinTemperaturecycle).
AstrongrelationexistsbetweenTkandLkinawaythat:IfTk,Lk0andifTk0,Lk.
(3)DuetoTAisappliedthroughaneighborhoodstructure,V,(PERTURBATIONfunctionmakesarandomperturbationfromSitogenerateaneighborhoodsolutionSj),themaximumnumberofdifferentsolutionsthatcanberejectedfromSiisthesizeofitsneighborhoods,|VSi|.
ThenthemaximumlengthofaMarkovchaininaTAalgorithmisthenumberofsamplesthatmustbetakeninordertoevaluateanexpectedfractionofdifferentsolutionsfromVSiatthefinaltemperatureTf,thisis:Lf=C|VSi|.
(4)whereCvariesfrom1C4.
6(explorationfrom63%until99%),LfisthelengthoftheMarkovchainatTf.
From(3),LkmustbeincrementedinasimilarbutinversewaythatTkisdecremented.
ThenforthegeometricreductioncoolingfunctionusedbyKirkpartick[6],andDueckandScheuer[9],Tk+1=Tk.
(5)theincrementalMarkovchainfunctionmustbe:Lk+1=Lk.
(6)where=exp((lnLf–lnLi)/n).
(7)Here,LiisthelengthoftheMarkovchainatTi,usuallyLi=1,andnisthenumberoftemperaturestepsfromTitoTfthrough(5).
Now,themaximumandminimumcostincrementproducedthroughtheneighborhoodstructureare:ZVmax=Max{COST(Sj)–COST(Si)}.
(8)ZVmin=Min{COST(Sj)–COST(Si)}.
(9)forallSjVSi,andforallSiSThenTiandTfmustbecalculatedas:Ti=ZVmax.
(10)TfZVmin.
(11)ThiswayofdeterminingtheinitialtemperatureenableTAtoacceptanypossibletransitionatthebeginningoftheprocess,sinceTiissetasthemaximumdeteriorationincostthatmaybeproducedthroughtheneighborhoodstructure.
Similarly,TfenablesTAtohavecontroloftheclimbingprobabilityuntilthealgorithmperformsagreedylocalsearch.
3Davis&PutnamMethodSatisfiablityProblem[12](orSAT)isveryimportantincomplexitytheory.
Letbeapropositionalformulalikeformula(12):F=F1&F2&…&Fn(12)whereeveryFiisadisjunction.
EveryFiisadisjunctionofpropositionalformulassuchasX1vX2v.
.
vXr.
EveryFiisaclauseandeveryXjisaliteral.
Everyliteralcantakeatruthvalue(0orfalse,1ortruth).
InSatisfiabilityproblemasetofvaluesfortheliteralsshouldbefound,insuchawaythattheevaluationof(12)betrue;otherwiseif(12)isnottrue,wesaythatFisunsatisfiable.
Besideswesaythat(12)isinConjunctiveNormalFormorCNF.
TheDavis&Putnammethodiswidelyregardedasoneofthebestdeterministicmethodsfordecidingthesatisfiability[12]ofasetofpropositionalclauses[10].
Itisalsoacompleteresolutionmethod.
Thisprocedurecallsitselfafterrewritingtheinputformulaaccordingtoanumberofrulesforgeneratingasmallerformulawiththesametruthvalue.
TherulesusedfortheDavis&Putnammethodare:Rule1:iftheinputformulahasnoclauses,thenitissatisfiableRule2:ifithasaclausewithnoliterals,itisunsatisfiableRule3:ifithasaclausewithexactlyoneliteral,thenmaketheliteraltrueandrewritetheformulaaccordinglyRule4:ifsomevariableappearonlypositivelyornegatively,thenpickonesuchvariableandassignavaluetoittomaketheliteraltrue,andrewritetheformulaaccordinglyIfnonerulecouldbeapplied,onepicksupanarbitraryvariableasabranchingpointandtwonewformulasarederivedbyassigning0and1tothisvariable.
Ifoneofthecallsreturnswiththepositiveanswertheinputissatisfiable;otherwise,itisunsatisfiable.
TheDavis&Putnamalgorithmisshownbelow:FunctionDAVIS-PUTNAM(Informula:clauseslist)BeginREDUCE(formula,vreduce)IfformulaisemptyThenReturnvreduceElseIfformulahasaclausewithnoliteralsThenReturnfailElseChoosealiteralVfromformulavaluation=DAVIS-PUTNAM(SUBSTITUTION(true,V,formula))Ifvaluation!
=failThenReturnADD(V=true,vreduce,valuation)valuation=DAVIS-PUTNAM(SUBSTITUTION(false,V,formula))Ifvaluation!
=failThenReturnADD(V=false,vreduce,valuation)ReturnfailEndifEndDAVIS-PUTNAMFunctionSUBSTITUTION(TF,V,formula)BeginForEachoneclauseCInformulaDoIf[CcontainVandTF=true]or[Ccontain~VandTF=false]ThendeleteCfromformulaElseIf[CcontainVandTF=false]or[Ccontain~VandTF=true]ThendeleteVfromCEndifEndforReturnformulaEnd_SUBSTITUTIONFunctionREDUCE(InOut:formula,vreduce)Beginvreduce=emptyWhileexistsclauseCInformulawithexactlyoneliteralLIfLispositivevariableVThenformula=SUBSTITUTION(true,V,formula)vreduce=CONS(V=true,vreduce)ElseIfLisnegativevariableVThenformula=SUBSTITUTION(false,V,formula)vreduce=CONS(V=false,vreduce)EndifEndwhileReturn(formula)End_REDUCETheDAVIS-PUTNAMfunctionisthemainfunctionanditselectsrandomlyaliteraltosetatrueagroupofvaluesinordertocreateunitaryclauses.
Ifthattruesetvaluesisnotthecorrectsolutionthecomplementsetoftruevaluesistried.
Ifthenewassignmentisneitherasatisfiablesolution,thentheformulaisunsatisfiable.
ThefunctionSUBTITUTIONmakesthepropagationofoneliteraloveralltheclausesinformula,deletingclauseswhereoccursthepositiveliteralLanditsvalueis1(true).
Thereforetheclauseswhere~Loccurscandeletethatliteral.
TheREDUCEfunctioncarriesoutthesearchofunitaryclauses,sothatitcanbepossiblepropagatethroughthefunctionSUBSTITUTION.
4GraphColoringthroughAcceptingandDavis&PutnamInformallycoloringagraphwithkcolorsorGraphk-ColorabilityProblem(GCP)isstatedasfollows:IsitpossibletoassignoneofkcolorstoeachnodeofagraphG=(V,E),suchthatnotwoadjacentnodesbeassignedthesamecolorIfanswerispositivewesaythatthegraphisk-colorableandkisthechromaticnumberx(G).
ItispossibletotransformGraphk-ColorabilityProblem(GCP)intoSatisfiabilityproblem(SAT);thatmeansthatforagivengraphG=(V,E)andanumberk,itispossibletoderiveaCNFformulaFsuchthatFissatisfiableonlyinthecasethatGisk-colorable.
TheformulationofGCPasSATismadeassigningXBooleanvariablesasfollow:1)TakeeverynodeandassignaBooleanvariableXijforeverynodeiandcolorj;thedisjunctionofallthesevariables.
Inthiswayeverynodewillhaveatleastonecolor.
Therefore,inthecaseoffigure1,wehavetheclauses:Node1:X11vX12vX13vX14Node2:X21vX22vX23vX24Node3:X31vX32vX33vX34Node4:X41vX42vX43vX442)Toavoidthefactthatanodehasmorethatonecolor,addtheformulaXij~Xik3)Inordertobesurethattwonodes(Vi,Vj)connectedwithanarchavedifferentcolors,addaclausesuchthatifVihascolork,Vjshouldnotbecolorwiththiscolor.
ThisclauseiswritingasXik~Xjk.
4)Inordertoknowwhichnodesareconnectedwithanedge,anadjacentmatrixAofthegraphisneeded;itselementsare:1ifiisconnectedwithjAij=0otherwiseFig.
1.
GraphcoloringexampleThereductionofagraphtotheConjunctiveNormalForm(CNF)generatessomanyclausesevenforsmallgraphs.
Forexample,forafullgraphwith7nodes(42edges),308clauseswith98literalscanbegenerated.
IfweuseDavis&Putnamalgorithmtocoloragraph,wecouldstartcoloringwithRcolors(thegraph'sdegreeorfromanumbergiven).
Ifitisnotpossibletocolorit,thenwecanincreaseRandtryagain.
Duetofindalargechromaticnumberx(G)isaveryhardtaskforacompletemethodasDavis&Putnam(itdemandsmanyresources),weneedanincompletemethodtohelpinthistask.
ForthisreasonwehavechosentheThresholdAcceptingmethod.
TAwillsearchthechromaticnumber,butasitisknownTAnotalwaysgettheoptimalsolution.
Bythisreason,thenumberfoundbyTAissendtoaDavis&Putnamprocedure,andthisonewillgettheoptimalsolution.
Thecompleteprocessisshowninthefigure2.
Fig.
2.
DescriptionofthecoloringprocessAnygraphcanbecoloredwithGmax+1colors,whereGmaxrepresentsthegraphdegree.
Forthisreason,TAwilltrycoloringwithGmaxcolors.
IfTAgetsasuccess,thenTAwilltrytocolorwithGmax-1,andsoon.
WhenTAfinishes,itsendstotheoutputtheminimumkofcolorsfounded.
Inothercase,whenTAcannotcolorwithGmaxcolors,thenitwillsendk=Gmax+1toDavis&Putnamprocedure.
Fig.
3.
BinarypartitionsDavis&Putnamwillattempttodecreasethevalueofkthroughbinarypartitions.
Thefirstattempt,Davis&Putnamwillchoosethenumberofcolorsgivenby(1+k)/2.
Ifthecoloringisright,itwillcolorwith(1+(1+k)/2)/2colors,i.
e.
,thelefthalf.
Otherwise,thealgorithmwillcolorwith((1+k)/2+k)/2colors,therighthalf.
ThisprocesscontinuesuntilDavis&Putnamcannotdecreasek.
So,thechromaticnumberwasfound.
Thissituationisshowninfigure2.
Thefigure3showsanexamplewhereTAfoundthenumbernineasitsbettersolutionanditissendtoDavis&Putnamprocedure.
WhenDavis&PutnamtakesthelastTAsolution,usingbinarypartitionsandotherrulestheoptimalsolutioniswaited.
Forexampleinthecaseofthefigure3,ifDavis&Putnamcannotcolorwithfivecolors,itmovestootheralternative,tryingwithsevencolors.
Finally,inthelastpartition,i.
e.
(7+9)/2,cannotcolorthegraphandsotheresultisachromaticnumberequaltonine.
5ConclusionInthispaperwepresentedanalgorithmbasedonThresholdAcceptingandDavis&Putnam,tosolvetheGraphk-ColorabilityProblem.
BecausethisproblemisanNP-hardproblemthereisnotaknowndeterministicefficient(polinomial)method.
Non-deterministicmethodsareingeneralmoreefficientbutanoptimalsolutionisnotguarantee.
Thismethodisanewalternativethatpromisestobemoreefficientthatthepreviousones.
Themaincontributionsofthispaperareenumeratedbelow.
1)Weproposedawaytotransformthegraphk-colorabilityproblemintoasatisfiabilityproblem.
2)InordertosolvetheformerproblemweproposedanewapproachwhichmakesuseofthethresholdacceptingandDavis&Putnamalgorithms.
3)Theresultingalgorithmiscompleteandusingitwecangetbetterresultsthatthewell-knownsimulatedannealingalgorithm.
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