University of Alberta

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1 TheGraphIsomorphismProblem University of Alberta ScottFortin by TechnicalReportTR96{20 July1996 DEPARTMENT OF COMPUTING SCIENCE The University of Alberta Edmonton, Alberta, Canada

2 theory,canbebroughttobearontheproblem,andyetonlyforspecial piesararepositionintheworldofcomplexitytheory,itisclearlyinnp butisnotknowntobeinpanditisnotknowntobenp-complete. Manysub-disciplinesofmathematics,suchastopologytheoryandgroup graphsthatlookdierentlyareactuallythesame.theproblemoccu- classesofgraphshavepolynomial-timealgorithmsbeendiscovered.incongruently,thisproblemseemsveryeasyinpractice.itisalmostalways Abstract Thegraphisomorphismproblemcanbeeasilystated:checktoseeiftwo Thispaperismostlyasurveyofrelatedworkinthegraphisomorphism implementationsexistsforapplicationdomainssuchasimageprocessing. trivialtochecktworandomgraphsforisomorphism,andfasthardware eld.weexaminetheproblemfrommanyangles,mirroringthemultifacetednatureoftheliterature.wesurveycomplexityresultsforthe haveconstantsthataresolargeastonegatetheirusefulnessinactual whichhaveknownpolynomial-timealgorithms.manyofthesealgorithms graphisomorphismproblem,anddiscusssomeoftheclassesofgraphs 1Introduction somegraphswhichare\hard"forpracticalalgorithms,andpresentexperimentalresultswhichdemonstrateaclassofgraphsonwhichpractical applications.thusanumberofpracticaltechniques,whichdonothavea algorithmsrunintodiculty. polynomialworstcase,arealsoexamined.weendthepaperbydiscussing doesthereexista1-to-1mappingoftheverticesinonegraphtotheverticesof theothersuchthatadjacencyispreserved?insymbols,giveng1andg2,does GraphIsomorphismDisease"[29].Workhascontinuedunabatedonthegraph isomorphismproblemhowever,duetothemanypracticalapplicationsofthe subject,somuchinfactthatin1977readandcorneilchristenedit\the problem,anditsuniquecomplexityproperties. Thegraphisomorphismproblemhasalonghistoryintheeldsofmathematics, thereexistanfsuchthat8a;b2v1;(a;b)2e1()(f(a);f(b))2e2.the chemistry,andcomputingscience.muchresearchhasbeendevotedtothis graphsinfigure1areexamplesofisomorphicgraphs. Thegraphisomorphismproblemcanbesimplystated:giventwographs, withintheconventionalcomplexityclasses,hasledtomanyresearchpapers;in FactoringisanotherproblemwhichisnotthoughttobeinPorNP-complete, isomorphism.thisambiguity,abouttheexactlocationofgraphisomorphism Section2wesurveysomeoftheseresults. howeverfactoringisknowntobeinco-np,whichisnotthecasewithgraph problemisknowntobeinnp,butisnotknowntobeinpornp-complete. knowinwhichcomplexityclasstheproblemlies.somewhatsurprisingly,the Asarstattemptatunderstandinggraphisomorphism,wemaywishto 1

3 isomorphismdiscusshowtobuildfast,practical,isomorphismcheckers.some systemtoquicklygiveeachgraphauniquename.thusmanypapersongraph routinelydealwithgraphswhichrepresentmolecularlinks,andwouldlikesome tionswhichdesireafastisomorphismalgorithm.forexample,organicchemists enoughtojustifyitsstudy,itturnsoutthattherearemanypracticalapplica- Whilethepositionofgraphisomorphismincomplexityanalysisismorethan Figure1:Isomorphicgraphs ofverticesanedgewithxedprobabilityp,arealmostalwayseasytosolveby Findinggraphswhichare\hard"forthestrongestpracticalalgorithmsisa non-trivialtask.graphswhicharerandomlygenerated,bymakingeachpair provablypolynomialworstcase,thesignicanceofthisfactisnotentirelyclear. asimplebacktrackroutinewithaverynaiveheuristic[16].insection4we worstcase.insection3weexamineindetailsomeofthepracticalresults. classofgraphs,butmostpracticalalgorithmsdonothaveapolynomialtime oftheseroutinescomefromspecialpolynomial-timealgorithmsforarestricted discusstheproblemofndinghardgraphsfortheisomorphismproblem,and giveexperimentalresultswhichdemonstrateaclassofdicultgraphs. Althoughnopracticalalgorithmwhichcanbeappliedtoallgraphshasa f(a)=aforalla2v.anautomorphismpartitionofagraphgisasequence theoreticconceptscannotbeavoided,andwegiveabriefsummaryhere.an ofdisjointsubsetsv1;:::;vk,suchthatforallpairsofverticesa;b2vithere copiesofg;i.e.itisamappingfbetweenpairsofverticessuchthat(a;b)2 existsanautomorphismwheref(a)=b1.thustheautomorphismpartition E()(f(a);f(b))2E.Atrivialautomorphismisanautomorphismwhere automorphismofagraphgcanbedenedasanisomorphismbetweentwo theorythatunderliesmanygraphisomorphismresults.however,somegroup Forthemostpart,wetrytohidethereaderfromthecomplicatedgroup dividesvintosetsconsistingofalltheverticesthatcanbemappedontoone another. 1Alternatively,wecouldsaythatViistheorbitofaintheautomorphismgroupofG. 2

4 notknowntobeineitherpornp-complete;further,giisnotknowntobein 2ComplexityResults worldofcomplexityanalysis.itisoneofthefewproblemswhichisinnpbutis atelyexponentialalgorithmforsolvingtheproblem[3].moderatelyexponen- tialmeansthatonaproblemofsizen,therunningtimetobeystherelation: problems. co-np.itiswellknownthatifp6=npthenthereexistsproblemswhichareof Thegraphisomorphismproblem(GI)occupiesanimportantpositioninthe nk<t<an,wherek;a>1arearbitraryrealnumbers.typicallythismeansthe intermediatestatus,andithasbeenhypothesizedthatgimaybeoneofthese exponentintherunningtimeisafunctionofnwhichgrowsslowerthann.for example,theoneupperboundforgeneralgraphisomorphismisexp(n1=2+o(1)) [24]Ḃabia,Erdos,andSelkowsshowedthatforalmostallgraphsX,isomorphism toanyothergraphycouldbecomputedinquadratictime[5].aswell,ithas OneearlyresultonthecomplexityofGIshowedthatthereisamoder- theoryproblems,namelyrandomgraphs,theisomorphismproblemisnotvery dicult. isomorphism[6].theseresultsshowthatunderametricappliedtoothergraph beenshownthatthereisalinearexpectedtimealgorithmforsolvinggraph determineclassesofgraphssuchthattheisomorphismproblemisinpwhen locationofgiinthesenewcomplexityclasses.athirdapproachistotryto eralizethenotionofnpintomoreesotericnotionsofcomplexity,andstudythe restrictedtographsofthattype. proachistodeneitasbeingitsowncomplexityclass,isomorphism-complete, andseewhichotherproblemsfallwithinitsclass.anotherapproachistogensearchershavetakenotherapproachestodeterminingitscomplexity.oneap- SinceGIhasresistedeortstoclassifyitineitherPorNP-complete,re- ofisomorphismsbetweentwographsisisomorphism-complete.thisresultis togi.mostsignicantly,heshowedthattheproblemofcountingthenumber 2.1IsomorphismComplete Mathon[25]showedthatanumberofproblemsarepolynomiallyequivalent problems.indeedevenforsomeproblemsinp,thecountingversionis(conjecturedtobe)muchharderthanthedecisionproblem. theproblemofdeterminingtheautomorphismpartitionofagraph.thegraph automorphismdecisionproblem(ga),whichasksifthereexistsanon-trivial numberofnon-trivialautomorphismsofagraph,whichisinturnequivalentto problems.heshowedthatgiisequivalenttotheproblemofcountingthe MathonalsodemonstratedtheclosetiesbetweenGIandautomorphism versionsofnp-completeproblemsaremuchharderthantheirrelateddecision furtherevidencethatgiisnotnp-complete,sinceitseemsthatthecounting 3

5 graphs,transitivelyorientablegraphs[10],andregulargraphs[9].theseresultsarenotterriblysurprisingsincemanyofthesegraphclassesexhibitalarge reductionintheotherdirectionisknowntoexist. phismproblemconnedtoanyofthefollowingtypesofgraphsisisomorphism- allyequivalenttogi.wesimplylist,withoutproof,thatthegraphisomor- Thetermequalityproblemwasalsoshowntobeisomorphism-completeby Manyrestrictedisomorphismproblemsarealsoknowntobepolynomicomplete:bipartitegraphs,linegraphs,rootedacyclicdigraphs[2],chordal automorphism,isaninterestingexception;whilegaisreducibletogi,nosuch dicult(seesection4). amountofregularity,whichisoneofthefactorsthatmakesgraphisomorphism Basin[7].Inthisproblem,wearegiventwotermswhichcanhavefunctions whichareassociative,commutative,orboth,andcanhavecommutativevariable itdoesnotoccurnaturallyasagraphtheoreticproblem.itlendscredenceto thenotionofisomorphism-completeasaseparateandusefulcomplexityclass. 2.2GeneralizationsofNP NPistypicallycharacterizedasthoseproblemswherethereisapolynomially problemisdierentfromthepreviousisomorphism-completeproblems,inthat bindingoperators.wethenmustdecideifthetwotermsareequal;aspecial checkableproofcerticate.itcanalsobethoughtofasaninteractionbetween caseofthisproblemisdecidingiftwopredicatecalculustermsareequal.this twoparticipants,anall-powerfulproverandaverier.theproverlooksatthe putstheverieracceptspositiveinstanceswithprobability3/4,andaccepts input,andcomputesaproofcerticatewhichhesendstotheverier;theverier thenchecksthecerticateinpolynomialtimeandeitheracceptsorrejectsthe input.pcanbethoughtofasthoseproblemswherethereisnocommunication accesstoastreamofprivaterandomnumbersandonlyrequirethatoverallin- betweentheproverandtheverier.aswell,welettheproverandverierhave negativeinstanceswithprobability1/4.ipisknowntobeequaltopspace, andisthusnotveryinteresting.howeverifwerestrictthenumberofmessages sages(forp)or1message(fornp),weallowapolynomialnumberofmessages toaconstantk,wegettheclassip(k),whichisknowntobeequaltoip(2)for allk>1andisnotknown(andbelievednot)tobeequaltopspace.inthis TheclassIPisageneralizationoftheabovescenario.Insteadofnomes- Goldwasser,Micali,andRacko[19]thatthegraphnon-isomorphismproblem isinip(1)andhencegiisinip(2).muchmoresurprisingistheresultof acceptingnegativeinstances0. wheretheprobabilityofacceptingpostiveinstancesis1andtheprobabilityof IP(2).NotethatPandNParespecialcasesofIP(0)andIP(1)respectively, notationwethushavethreeclasseswhichmaynotbeequal,ip(0),ip(1),and Comingbacktographisomorphism,wehavethatsinceGIisinNP,GI 4

6 proofthatgi2ip(2) anddemonstratesthebasicideaofaninteractiveproof: (whichisnotknowntobeinnp)isinip(2).thealgorithmisquitesimple, 3.I 6.ifa=iandb=jtheverieraccepts,otherwiseitrejects 5.theproverreturns2numbersi;jsuchthatGiisisomorphictoI, 4.theveriersendsI;Jtotheproverinparallel 2.theverierpicks(atrandom)a;b2f1;2g 1.lettheinputgraphsbeG1andG2 alwaysaccept.iftheyareisomorphic,theproverhasnowayto knowwhichgraphiwasderivedfrom,andsowithprobability1/2 IfG1andG2arenotisomorphic,thenclearlytheverierwill andgjisisomorphictoj someisomorphismofga,j someisomorphismofgb complete,givesacomplexitydiagramlikethatoffigure2.thissignicance ofthisclassicationisnotsolelytheincreasedperspectiveitgivesonthegi problem,butithasalsobeenusedtogivestrongevidencethatgiisnotnpcomplete. Thisresult,combinedwiththelackofresultsplacingGIinPorNP- probability1/2thatj6=b.thusifg1andg2areisomorphic,the itwillreturnisuchthati6=a.similarlyforgraphj,thereis verieracceptswithprobability1=4.2 IP[2] PSPACE PH1PH2:::,basedonexistentialquantiers,universalquantiers,and Thepolynomialhierarchy(PH)isahierarchyofcomplexityclassesPH0 Figure2:ComplexityofGIandGI 5 NP-complete co-np co-np-complete NP P

7 andzachos[11]andschoning[31]prove(indierentways)thismainresultof graphisomorphismstructuralcomplexity: andph1=np.wewillnotdiscussthedetailsofph,excepttosaythatitis generallybelievedtobeaproperhierarchy[23].thatistosay,itisbelieved theredoesnotexistanisuchthatforji,phi=phj.boppana,hastad polynomialtimealgorithms.ittooisageneralizationofnp,i.e.ph0=p IfthegraphisomorphismproblemisNP-complete,thepolynomial inagraphwithaspeciallabelededge.ahighleveldescriptionofthealgorithm thepaper,giisreducedtotheproblemofndingthenumberofautomorphisms 2.3GraphswhereGIisinP TherstmajorresultinthiseldwasapaperbyLuksin1978[24].Heshowed thatforgraphswhichhaveboundeddegree(i.e.themaximumdegreesome constantk),thereexistsapolynomialtimealgorithmtocheckisomorphism.in hierarchycollapsestoph2=ip(2) intox1;:::;xn?1wherexiistheinducedsubgraphonthoseverticeswhich proceedsinthreesteps.firstanedgeeischosen,andthegraphispartitioned andluksclaimsthatusingmoresophisticatedtechniquestherunningtimeis stepisalwayspossibleandrunsinpolynomialtime.thealgorithmpresented inthepaperwasnotmeanttogiveatightupperboundontherunningtime, ino(ncklogk),wherec>1isaconstantandkisthemaximumdegree.itis forallgroups.usingpowerfulideasfromgrouptheory,luksprovedthateach itssetofneighboursinxi+1,andverticeswiththesamemappingaregrouped together.finallyeachofthesegroupsisdecomposedintoprimitivegroups wherethenumberofautomorphismsisknown,andthesenumbersaresummed areistepsorfewerawayfrome.next,ineveryxi,eachvertexismappedto clearthatevenifthemaximumdegreeisonly10,thistechniquedoesnotyield apracticalalgorithm. inpolynomialtime.thisalgorithmisnotverypractical,asitsrunningtimeis phism,onesuchpropertyisthegenusofagraph.ifwewishtodraw(more isdenedasasequenceofnumbersg1;g2;:::suchthatgiisthenumberofways wexk,thenthosegraphswhichhavegk>0canbecheckedforisomorphism commonlyembed)agraphonsomesurfacesuchthatnotwolinescross,wemay todrawagraphonasurfaceofgenusi.filottiandmayer[18]showthatif needasurfacewithafairlycomplicatedstructure.thegenusisameasureof howcomplicatedsuchasurfacemustbe;andthegenusdistributionofagraph Othergraphpropertieshavebeenstudiedwithrespecttographisomor- constantinthisalgorithmis(512k3)!whichisverylargenomatterwhatthe O(nck)wherec>300isaconstant.Chenobtainsabetterresult[13],showing thatifyoutaketheaverageofthegi's,andrequirethattobeboundedbya valueofk.planargraphsareinsomesenseaspecialcaseofthisresult,asthey constantk,thenthereisalineartimeisomorphismtest.however,thehidden 6

8 isomorphisminlineartime,althoughtheiralgorithmalsohasalargeconstant whichmakesitunsuitableinpractice. rstshownbyhopcroftandwong[20]thatplanargraphscouldbecheckedfor canbedrawnonaplane(orasphere)andhencehavegk>0forallk1.itwas thesegraphs,wheremisthenumberofedges,andnisthenumberofvertices. Thisresultisinterestinginthatitdoesnotrequiresomeexplicitparameterto beconstant,andseemstoapplytoalargeandpracticalgroupofgraphs. Hsu[21]hasshownthatthereisaO(mn)algorithmfortestingisomorphismof andjoiningtwoverticesiftheirarcsoverlap;asubclassofcirculararcgraphs createdbydrawinganumberofarcsonacircle,creatingavertexforeacharc, istheintervalgraphs,whereour\arcs"areconnedtotherealnumberline. circular-arcgraphs[21].circulararcgraphsarethosegraphswhichcanbe AnotherlargeclassofgraphsoverwhichGIcanbesolvedecientlyisthe phismcanbesolvedinpolynomialtime.forexample,cogisandguinaldo isomorphismproblemswhichareinp. permutationgraphs[15],chordal(6,3)graphs[4]andpartialk-trees[8]allhave techniqueforknowledgerepresentation.othergraphclassessuchastrees[2], [14]showthatisomorphismforconceptualgraphswhichhavealinearorderingfunctioncanbesolvedinpolynomialtime.Thisisparticularlyofinterest tothearticialintelligencecommunity,whereconceptualgraphsareusedasa Theseresultstakentogetherprovideaninterestingcounterpointtothediscussionofrandomgraphs.Itseemsthattheisomorphismproblemiseasyfor graphswhicharenotrelatedinanyway(randomgraphs)andforgraphswhich exploredmorefullyinsection4. 3PracticalIsomorphismAlgorithms Oneofthefactorscontributingtothelargeamountofworkonthegraphisomor- approachistotakethetwographswhicharetobecompared,andtrytodi- considered. rectlyndanisomorphismbetweenthem.thishastheadvantagethatifthere phismproblemisundoubtablythemanypracticalapplicationsofthetechnique. Therearetwobroadapproachestosolvingtheisomorphismproblem.Therst tionc(g)whichreturnsacanonicallabel;whereacanonicallabelmeansthat aremanyisomorphismsbetweentwographs,thenonlytherstoneneedbe theisomorphismproblem,butispossiblyofindependentinterest.forexample, C(G)=C(H)iGandHareisomorphic.Suchalabelingnotonlysolves organicchemiststypicallydealwith\graphs"suchastheoneiffigure3.itisa fundamentalproblemintheworldofchemicaldocumentationtogiveeachsuch structureauniquename. ThesecondapproachistotakeasinglegraphGandcomputesomefunc- Thereareseveralotherrestrictedclassesofgraphsforwhichgraphisomor- arestructurallyrelated(circulararcgraphsforexample).thisnotionwillbe 7

9 H H O werepresentgraphsbyanadjacencymatrixm,wheremij=1if(i;j)isan Figure3:Atypicalorganicchemistrygraph C C H C C C C C C H inpractice. edge,and0otherwise.wecanconstructalabelforgbyconcatenatingthe isisomorphism-complete,sothismethodhastobeslightlymodiedtobeused labelforgbylookingatthelabel(asdenedabove)ofeveryautomorphismof G,andreturningthesmallestone.OfcoursendingeveryautomorphismofG rowsofmtoformannbinarynumber.wecanthencomputeacanonical Itisnothardtodescribeanaivecanonicallabelingsystem.Supposethat C C O vertexinvariantswhichareanimportantpracticaltoolinsolvingisomorphism traditionalandspecialcaseapproachestographisomorphism.wethenlookat Therestofthissectionisorganizedasfollows.Werstbrieylookatnon- H H 3.1Non-traditionalApproachestoGI Therehasbeensomeworkonnon-traditionalapproachestothegraphisomorphism.McGregor[26]discussesusingarcandpathconsistencyalgorithmson problems,andnallywelookatalgorithmsforsolvingthegraphisomorphism problemusingeachofthetwoapproachesoutlinedabove. isomorphismproblemcouldverypossiblybeinp.theresultsofthepaper previous(relativelysimple)backtrackingalgorithms. seemtottheintuitivenotions,whereonlyaslightincreasewasobservedover astheconstraintsatisfactionproblemisknowntobenp-hard,andthegraph tunedconstraintalgorithms.thisapproachdoesnotseemintuitivelyinviting, phismproblemintoaconstraintsatisfactionproblem,andthenusesspecially theisomorphismproblem.thealgorithminthepaperconvertsagraphisomor- Agusa,Fujita,Yamashita,andAe[1]presentaHopeldnetworkforsolving 8

10 ting.thereasonforthisisthefactthatthemodelrestrictsthemaximum numberofverticesallowedinagraphto12.clearlythispaperprovesthat howeverthisapproachcertainlycannotbeusedunmodiedinapracticalset- neuralnetworksolutionstotheisomorphismproblems(forsmallgraphs)do exist,althoughitseemstodolittletoadvancethetheoryorpracticeofgraph result.inpractice,wemaybewillingtolivewithasmalldegreeofuncertainty, isomorphismproblems.itisuncleartowhatproblemdomainthistypeofsolutionissuited.theoretically,thisapproachcannotproveapolynomialworst case,sinceitisnotpossibletoguaranteethatitwillalwaysreturnthecorrect isomorphism. bydemarchi,masera,andpiccinin[17],whoimplementedplanarisomorphism bygivingafast,parallelplanarisomorphismalgorithm[22].giventhereal timeconstraintsofimageprocessing,thisalgorithmwastakenonestepfurther AswediscussedinSection2,HopcroftandWongshowedplanarisomorphism checkingcanbesolvedinlineartime[20].jajaandkosarajuextendthisresult planargraphswhichrepresenttheinterrelationshipsbetweenobjectsinascene. checkinginavlsiprocessorarray.whiletheseresultshavelittlebearing isomorphismchecking.thegraphswhichariseinthisapplicationdomainare Imageprocessingrankswithchemistryasalargeapplicationdomainfor ontheisomorphismproblemasawhole,theydoshowthatsomepractical implementationshavebecomeveryecient. 3.2VertexInvariants someisomorphism,thencertainlytheymusthavethesamedegree.notethat Oneverypracticaltechnique,whichisindependentofthealgorithmusedto notnecessarilyanisomorphismwhichmapsthetwoverticesontooneanother. avertexinvariantisthedegreeofavertex:ifvandv0maptoeachotherunder theconversedoesnothold,iftwoverticeshavethesamedegree,thenthereis Infact,ifthereisapolynomiallycomputablevertexinvariantforwhichthe someisomorphismthatmapsvontov0theni(v)=i(v0).thetypicalexampleof actuallysolvetheisomorphismproblem,istheuseofvertexinvariants.a problem. vertexinvariantissomenumberi(v)assignedtoavertexsuchthatifthereis conversedoeshold,thenthereisapolynomialtimesolutiontotheisomorphism adjacencytriangles Belowwelistsomeinvariantswhicharepresentinanisomorphismprogram twopaths callednauty[27]. Therearemanyvertexinvariantswhichhavebeenproposedintheliterature. 2assignanumbertovbasedonthesizeofthecommonneighbourhoodof assignanumbertovbasedontheverticesreachablealongapathoflength 9

11 k-cliques independentk-sets thoseneighboursofvwhoareadjacent distances assignanumbertovbasedonthenumberofdierentcliquesofsizek ofsizekthatcontainv assignanumbertovbasedonthenumberofdierentindependentssets space.morepractically,vertexinvariantsaretypicallyusedbycombiningthe showthatusingvertexinvariantsmaynotresultinanydecreaseofthesearch resultsofmanyinvariants.forexample,ifavertexhasdegree3,isin2cliquesof size4,andcanreach9verticesonpathsoflength2,thenacompositeinvariant mightbe329.infact,thisabilitytocomposeinvariantshasledtothecase Theuseofinvariantsissomethingofablackart.CorneilandKirkpatrick[16] assignanumbertovbasedonthenumberofverticesateachdistance wherepracticalalgorithmsalwaysusedegreeasthe\base"invariant,andother 1;:::;nfromv thebestforaparticulargraph.instead,whatistypicallydoneistoleavethe decisionofifandwhentouseavertexinvariantuptotheuser(exceptfordegree variantsgetappendedtothebase.theseotherinvariants,whiletheycanbe invariants[27],anditisverydiculttodeterminewhich(ifany)invariantis whichisalwaysused).thisassumesthatiftheuserhappenstocomeacross computedinpolynomialtime2,canbequiteexpensivetocalculate,compared degreeinvariantissucientformostgraphsencounteredinpractice,andthe anumberofveryhardgraphs,hewouldthentakethetimetoexperimentally determinewhatinvarianttouse.anotherwayofjustifyingthis,isthatthe canbedirectlysolvedinlesstimethanittakestocalculatethemorepowerful tothetotalexecutiontime.itturnsoutthatmanyisomorphismproblems Perhapsthemostnaturalwaytotryandsolvethegraphisomorphismproblem istotryanddirectlydiscoveramappingbetweentheverticesoftwographsg1 3.3SolvingtheIsomorphismProblemDirectly useronlyhastointerveneifthegraphhasaveryregularstructure. areclearlynotofpracticaluse [30].WestartbydeningG1(k)asthesubgraphofG1inducedonthevertices andg2.thealgorithmpresentedinthissectionisanadaptationoftheonein 1;:::;k.ThebasicideaistobuildanisomorphismfromG1(k)toasubgraph anunusedvertexofg2. ofg2,andattempttoextendthistoanisomorphismong1(k+1)byadding 2Actuallyinvariantshavebeenproposedwhichdonotruninpolynomiallytime,butthey 10

12 Themainroutineofthealgorithmlookssomethinglike: ProcedureMain: Output:arrayfsuchthati!fiisanisomorphismbetweenG1andG2, Input:graphsG1andG2 fori ornoneifnosuchfexists inv1i 1;:::;ndo elsereturnnone ifisomorph(emptyset,1,f)then reordertheverticesofg1(andreorderinv1aswell) endfor ifinv1sortedinascendingorder6=inv2sortedinascendingorder returnf inv2i somevertexinvariantonthei-thvertexofg1 somevertexinvariantonthei-thvertexofg2 optimizationistoreordertheverticesofg1,withthehopethattherewillnotbe methodsmaybeused,forexampletheverticesing1maybesimplysorted used.thecalculationofinv1andinv2allowsustorejectmanynon-isomorphic manyverticesing2thatcanmaptotherstfewverticesg1.severalordering graphs,providingourchosenvertexinvariantispowerfulenough.thesecond basedonthenumberofverticeswhichshareacommoninvariantvalue(the ischosenandadepth-rstsearchrootedatthatvertexisusedtore-orderthe invariantmultiplicity),orperhapsavertexwithminimuminvariantmultiplicity Wecanseethatthisisoneoftheplacesinthealgorithmwhereheuristicsare vertices.thesimplesortensuresusthatasweenlargetheisomorphismthere ofg2inducedbyscanbeextendedtoacompleteisomorphismong1andg2. depthrstsearchapproachisattractiveinthatadjacentverticesareplacednear input.itreturnstrueiftheisomorphismbetweeng1(k?1)andthesubgraph isaminimumnumberofcandidatestoextendthesolutionateachstep.the oneanother,andthuswewouldhopethatifwemakinganincorrectmapping decision,wewouldquicklygetacontradiction. TheISOMORPHproceduretakesasetS,andnumberk,andanarrayfas ProcedureISOMORPH: Input:setS,integerk,andanarrayf Output:TRUEiffcanbeextendedtotheentireinputgraphs ifk=n+1thenreturntrue foreachj2(v2=s) ifisomorph(k+1;s[fjg)then if(inv1k6=inv2j)_:canmatch(k;j;f) gotonextloopiteration j 11

13 therealreadyexistsanisomorphismbetweentherstkverticesofg1and theunmappedverticesing2andtriestomapthemtothenextvertexing1.it theverticesins.ifithasalreadymappedallpossibleverticesitimmediately returnstrue,andanisomorphismfhasbeenfound.otherwise,itlooksatall Itisnottoohardtoseewhatthisroutineisdoing.Whenitiscalled, endfor returnfalse returntrue usesthevertexinvariantheuristicsothatitwillnevertryamappingwherethe invariants,inwhichcasewecouldusethepartialmappingftohelpuscreate shownasacachedvalueonthisexample,itispossibletodynamicallycalculate invariantisdierent.thisisthemeansbywhichmuchofthesearchtreecanbe amorediscriminatinginvariant. pruned,providedwehaveagoodinvariant.notethatalthoughthisinvariantis Inthissectionwepresentacanonicallabelingalgorithm,whichisactuallyahigh secondone. 3.4DeterminingaCanonicalLabeling andcheckingthe(i;k)isanedgeoftherstgraphi(fi;j)isanedgeinthe simplyscanningtheisomorphismalreadydiscovered(whichiscontainedinf) thegivenmappingwouldnotconictwithpreviousmappings.thisworksby TheonlyfunctionlefttodescribeisCANMATCH,whichreturnstrueif leveldescriptionofthealgorithmpresentedbymckaycallednauty[28,27]. Thisisaverypowerfulalgorithm,andiscurrentlythepreferredmethodfor solvingthegraphisomorphismproblem.thereasonthismethodoutperforms (ingeneral)thealgorithmintheprevioussection,isthatbyconcentratingon broughttobearontheproblem,signicantlydecreasingtherunningtime.as onegraphatatime,powerfulideasfromtherealmofgrouptheorycanbe well,thisalgorithmprovidesmoreinformationthanthepreviousalgorithm, Centraltotheoperationofnautyistheideaofanorderedpartition.Apartition listing(forexample)alloftheautomorphismsofagraph. dividestheverticesofagraphintonon-empty,disjointsubsetsofv(called readerto[28]formoredetails. Partitions reader.insteadwewillconcentrateonthemainideas,andrefertheinterested itwouldrequirealargeamountofgrouptheorybackgroundonthepartofthe Wewillnotgiveasdetailedanalgorithmasthatintheprevioussection,as theirroleasleavesofthesearchtree cells):v1;v2;:::;vm.apartitionwithonlysingletonsets,willbecalleda leafpartition3.akeypointisthatwhenwendaleafpartition,itdenesa 3Innautythesearereferredtoasdiscretepartitions,howeverthetermleafpartitionhilights 12

14 relabelingofthegraphwherethevertexincellvigetsrelabeledtoi. b a c a b c c a b a c a c b c a b aregivenfortheverticesinthegraph,andassociatedwitheachpartitionis b a theadjacentmatrixwheretheorderoftheverticesintherowsandcolumns c b correspondstotheorderoftheverticesinthepartitions.automorphismsare recognizedbyndingtwodistinctleafpartitionswhich(afterrelabeling)give andthusdeneanautomorphism. risetothesamegraph.intheexamplejcjajbjandjajcjbjgivethesamematrices Forexample,considerthegraphinFigure4.Threedierentleafpartitions Figure4:Discoveringanautomorphismbycomparingleafpartitions a c b discovered automorphism c -> a, a -> c, b -> b a a c b invariantscanbeused.theinvariantiscomputedforeachvertexovertheentire partitionandgeneratingthechildrenofapartition.renementiswherevertex c graphtocreateaninitialpartition,andthentheinvariantrestrictedtothecells Therearetwomajoroperationsthatnautyperformsonpartitions,reninga b returnsthenumberofneighboursofvthatareinthesets.wecouldthen ofthepartitiontotrytofurtherdistinguishthem.forexampleassumed(v;s) denearenementoperationas: 1) 2)selectaVi2suchthatVihasmorethanoneelement 3)foreachv2Vi,computeasequenceav=(d(v;V1);:::;d(v;Vm)) theinitialpartitionv1;:::;vmwhereforallverticesx;y2vi wehaved(x;v)=d(y;v) 13

15 Inactuality,wedonotjustselectasingleVibutstartingatV1weloopthrough allthevi(includingonescreatedafterasplit)untilwecannotsplitanymore partitionwhichisv1;:::;vi?1;fvg;vi=fvg;vi+1;:::;vm.thatistosay,we hasmorethanonemember,andthenforeachvertexv2viwecreateachild cellsẇegeneratethechildrenofapartitionbychoosingtherstcellviwhich 4)splitViintoanumberofsubsets,sotheverticesineachsubsethave createanewchildforvertexvbysplittingthecellviintotwocells,onewhich thesamevalueforav eachpartitionbeforeexpandingitschildren. rstsearchofthespaceofpartitions,withtheaddedoptimizationthatwerene operations,whichwewilltalkaboutinthenextsection.itisbasicallyadepth BasicAlgorithm Hereisthebasicalgorithmusedbynauty.Wehaveomittedsomeofthepruning containsonlyvandonewhichisviwithoutv. Output:acanonicalgraphC Input:agraphG while(sisnotempty) S thepartitioncontainingasinglecellv stackcontaining endwhile returnc if(xisaleafpartition) elserene(x) endifappendthechildrenofxtos x popthetopofthestacks update(c;x) CanonicalLabeling Conceptually,nautyexamineseveryautomorphismofagraphandcomputes willhashtodierentvalues. acanonicallabel.thislabelissimplytheadjacencymatrixofthe\smallest" automorphism.theschemeusedtodenethesmallestautomorphismanextensionoftheschemewediscussedpreviously,wheregraphsareorderedbasedon thebinarynumberthatiscreatingbyconcatenatingtherowsoftheadjacency input,andcomputesahashfunctioninthehopethatdierentautomorphism matrix.themainadditionisanindicatorfunction,whichtakesapartitionas 14

16 xbystartingattherootofthesearchtreeandtransversingdowntox,at eachpointtheresultoftheindicatorfunctionisusedtodeneamemberof thesequence.thusaleafpartitionisdenedassmallerthanif<4, andtheequivalentbinarynumberofeachautomorphismisusedtobreakties. McKayprovesthatacanonicallabelingdenedinthisfashionisthesamefor anytwoisomorphicgraphs[28]. posewehavesomepartitionxwhichisthesmallestpartitiondiscoveredsofar. Thusifwereachaleafpartitionxinoursearchtree,wedeneasequence Thenwecancomparesomenodevtox,andifvisbiggerwecanprunev. Bydeningtheindicatorfunctionoverallancestorsofx,wemaybeableto prunenodeswhicharenotleafnodes,sincetheymaybebiggerthantheancestorsofx.notethatifwesimplyorderautomorphismsbasedontheirbinary representation,thenwecanonlyprunenodeswhichareleafpartitions. Itiseasytoseehowthisinformationcouldbeusedtoprunethetree.Sup- iftwodierentleafpartitionresultinthesameadjacencymatrixafterrelabeling thevertices.nautystorestwographs,whichitusestocheckforautomorphisms. Thereisonlyonelastmaintechniqueemployedbynauty,therecognitionand UsingAutomorphisms Onegraphcorrespondstothe\smallest"graphfoundsofar,asdenedabove, useofautomorphisms.asmentionedearlier,nautyrecognizesanautomorphism graphiscomparedtothestoredcopies,andiftheyhaveexactlythesame andwilleventuallybecomethecanonicallabel;theothergraphcorrespondsto therstleafnodediscovered.whenanewleafnodeisreached,itsrelabeled adjacencymatrix,thenanautomorphismhasbeenfound.atthesametime, ifnautydiscoversthenewgraphissmallerthansmallestsofar,thenthenew graphbecomesthenewcanonicalgraph.notethatthiswillnotdiscoverall isatradeo morestoredgraphsmeansmorematches,butitalsomeansthe timetocheckeachgraphisincreased.inadditiontotheseautomorphisms,an automorphismcansometimebeinferredbythestructureofapartition;however automorphisms,sinceweareonlystoringtwographstocompareagainst.there thistechniqueislessimportantandwedonotdiscussitfurther. searchspace.oneusefulresultisthatifthereisanautomorphismbetweenaleaf partitionandsomestoredpartition,thennautycanjumpallthewayback another.thusisanapproximationoftheautomorphismpartition,andgets updatedwheneveranewautomorphismisfound.thesignicanceofcomes inifnautyhasdiscoveredsomeautomorphismwhichmapsthetwoontoone totherstcommonancestorofandandcontinuefromthere.additionally, nautymaintainsaspecialpartition,wheretwoverticesareinthesamecell intoplaywhenwegeneratethechildrenofsomepartition.recallthatthis Oncenautyhasfoundanautomorphism,itputsittoworktotrytoprunethe wordsareordered 4Thetwosequencesarecomparedinlexographicorder,i.e.inthesamemannerasEnglish 15

17 isdonebychoosingacellw2,andgeneratinganewchildforeveryelement ofw.given,wenowonlycreateachildforeachelementofwwhichisina uniquecellof.thusaswendmoreandmoreautomorphisms,lessandless childrengetexpandedfromeachnode.theproofthatthesepruningsarevalid dependsonsomefurtherideasaboutpartitionsandgrouptheory,whichwewill morphism.afundamentalweaknessoftheseapproachesisthefactthatonce grouptheoryideastohelpprunethesearchspace.insomesense,thesemantics theproblemistransformedintothatparadigm,itisnolongerpossibletouse 3.5Discussion Practicalexperienceseemstoindicatethatothergeneralparadigms,suchas constraintsatisfactionandneuralnetworks,arenotveryusefulforgraphiso- notdevelophere.theinterestedreadercansee[28]. ofthedataisbeinglostinthetranslation. itisnotdiculttospeculatewhynautydoesbetter.bothprogramsmakeuse directapproachtoisomorphism.thedierencesherearenotasconclusive,but ofvertexinvariants,nautyusestheminitsrenementprocess,andthedirectalgorithmusesthemwhendecidingwhethertoextendanisomorphism.themain dierencebetweenthetwotechniquesseemstobetheiruseofpartialresults. towork,pruningawaysectionsofthesearchtree.thuswhennautyexplores Whenevernautyndsanautomorphism,itimmediatelyputstheautomorphism aportionofthesearchtree,italwaysattemptstousetheseresults.contrast Anotherinterestingnoteisthesuccessofaprogramlikenauty,overamore thiswiththedirectscheme.supposewehaveextendedtheisomorphismtoall inthe\almostisomorphism"toprunesubsequentattempts,andthusweare isanumberoflevelsback.thereisnoattempttousetheinformationgained essentiallydiscardingtheknowledgewelearnedinthatportionofthesearch. butonevertex,andthenrealizethatwehavetobacktracktoapointwhich 4HardGraphs inrealtime.thelistofpropertiesthatour\hard"graphscannothavegoeson wonderifthishasanypracticalsignicance.putanotherway,wemaywishto knowifthereareanyreally\hard"graphs,andifthereare,getsomehandleon whatmakesthemdicult. WhilethegraphisomorphismproblemisnotknowntobeinP,onemight andon,andevenifourgraphdoesnothaveoneofthoseproperties,practical weconneourselvestorandomgraphs,thentheisomorphismproblemisalmost alwaysveryeasy.ifweinsteadfocusonplanargraphs,thereexistsaparallel hardwareimplementationthatwillndtheisomorphismsofverylargegraphs Thesearchforhardgraphsisnotatrivialone.Aswehavealreadyseen,if 16

18 rithmforaspecialsubclassofgraphs,itisoftennon-trivialtodetermineifan inputgraphbelongstosuchasubclass.finally,thearrayofheuristicsavailable forthealgorithmtodecideifandwhentoapplyeachone. topracticalalgorithmsisindeedlarge,butthatalsoservestomakeitnon-trivial thatmakethemprohibitiveforactualuse.further,evenifthereisafastalgo- theisomorphismproblem. rithmsthatsolveacertainclassofgraphsinpolynomialtimehaveconstants algorithmshaveadazzlingarrayofheuristicsthatcanbeusedtoquicklysolve section,andattempttondgraphswhicharediculttoprocess.ourapproach Inthissectionwelookatthenautyprogramintroducedintheprevious Practicalalgorithmsdonotgetothateasilyhowever.Manyofthealgo- asameanstopruneoutobviousnon-isomorphisms(ornon-automorphismsin isnottodirectlyconstructthesehardgraphs,butinsteadtotransformtypes nauty'scase).thustherstdecisionwecanmakeistorestrictourselvesto wheretolookforhardgraphs.aswenoted,allpracticalalgorithmsusedegree Ourpreviousdiscussionoftheisomorphismalgorithmsgivesussomehintsasto thepowerandlimitationsofpracticalgraphisomorphism. 4.1FindingHardGraphs of\easy"graphsintohardones.indoingsowegainabetterunderstandingof wishtoincreasethisnotionofregularity,byrequiringthatrangeinsizesofthe commonneighbourhoodofanytwoverticesbeverysmall. regulargraphs.agraphisregularifeveryvertexhasthesamedegree.wemay thereisnoknownalgorithmtouniformlygeneraterandomgraphswiththe withourdesiredregularityproperties. aboveproperties.instead,wehavetotrytond\naturallyoccurring"graphs lentgraphs.thesegraphsaregeneratedbycreatingmany(conceptually,in- nitelymany)ringsofregularp-gons,anddeningafunctionwhichmapsthese polygonsontooneanother{forexample,startatanyvertexandmapthethree polygonswhichshareittothethreepolygonsthatsharethevertexthatis14 Onesuchsourceofregulargraphsisthep-regularmapembeddingsoftriva- Unfortunately,oncewemakethesedecisionswereachapracticaldiculty; verticestotheright.afterperformingthismapping,eithertheringsofpolygonsremaininnite(oratleastbeyondthesizethatcanbehandledbythe computer'smemory),orallpolygonsaremappedtothesinglepolygonatthe center,orthereissomenitemapofpolygonswhichisproduced.theinterested nautycanassignthemacanonicallabel.itturnsoutthesegraphsareinsome sensetooregular,thenumberofautomorphismsisverylarge,whichcanbe readerisreferredto[12]formoredetailsonthisclassofgraphs.thelatercase exploitedtoprunethesearchspace.thekeyideainmakingthesegraphshard Oncewehavethesegraphsinhand,wecantesthoweasilyprogramslike isofparticularinteresttous,sinceitgeneratesahighlysymmetricp-regular graph. 17

19 theirendpoints.theedgesarechosenatrandom,butareconstrainedsothat istodecreasetheamountofself-similaritybychoosingtwoedgesandswapping whentheyareswapped,thedegreeofeachvertexdoesnotchange.asthe experimentalresultsinthenextsectionshow,doingonlyoneoftheseswapscan 4.2ExperimentalResults Fourgraphswerechosentoexaminetheeectofrandomswapsonthediculty maketheproblemseveralhundredtimesharderfornauty. ofthegraphthatresults.thegraphsareentitledv190,v348,v486,and v864,wherethenumberineachcaserepresentsthenumberofvertices.all programwrittenbyjoeculberson[12]basedontheconstructionmentionedin graphswereregulargraphswithdegree8,andwerecreatedusingagenerator theprevioussection. time vs number of swaps 3 v measurement,excepttomentionthattheywereverylowinallbutonecase, 1 whichwewilldiscussshortly. timeofnautyoverthosegraphs.wedonotexplicitlylistthevariancesforeach 10graphswiththerequirednumberofswaps,andthenaveragingtherunning 12,16,and32randomswaps.Eachmeasurementwasobtainedbycreating Figures5through8showauniformpictureoftheeectofrandomswaps. Foreachgraphwemademeasurementsfor0,1,2,3,4,5,6,7,8,10, Figure5:Resultsforv time in seconds

20 time vs number of swaps 30 v Ineachcasenautywasveryfastiftherewasnorandomswaps,andthenmuch slowerwhenrandomswapswereintroduced.further,asmoreswapswereintroduced,nautybegantodecreaseitsrunningtime mayexplainthelargeincreaseintimebetween0and1swaps,isthenumberof automorphismsintheresultinggraph.whentherewerenoswaps,thenumbers Figure6:Resultsforv348 ofautomorphismsofv190,v348,v486,andv864were3420,6144,7776,and respectively.Basedonourpreviousanalysisofthenautyalgorithm,it isnothardtoimaginethattheselargegroupsizeshelpnautytoaggressively 0 prunethesearchspace.afterdoing1ormorerandomswaps,thenumberof Thereareafewplausibleexplanationsforthisbehaviour.Onefactorwhich automorphismswasalmostalways1;outof520graphsthatcontainedrandom number random swaps automorphismgroupofsize2,andthesewereabletocompleteinapproximate oftheonemeasurementwithlargevariance:itcontainedtwographswithan swaps,only5hadanautomorphismgroupofsize2.infactthisisthecause 2/3theamountoftimeastheotherruns.Clearlyinthegraphswithrandom swaps,nautycannotuseautomorphismstohelpprunethesearchtree. increased,asallofthesegraphshadtheapproximatelythesamenumberof eectively.witheachrandomswapweessentially\break"thestructureofthe thegraphswithmanyswapsbecauseitisusingitsrenementproceduremore automorphisms(1or2).intuitivelywecanarguethatnautyisspeedingupon Thisdoesnotexplainthefactthetimedecreasedasthenumberofswaps 19 time in seconds

21 time in seconds time vs number of swaps v486 Figure7:Resultsforv number of random swaps time in seconds time vs number of swaps 600 v Figure8:Resultsforv number of random swaps

22 manyconclusionscanbemadeabouttheseresults,excepttosaythatmore graph,andthuswithmanyswapsthenumberofbreaksincrease,andhence asitrelatedtographsize.foreachgraphweplottheblowupratior=(time withoneswap)/(timewithnoswaps).forthistestweincludedanumberof byinterlockinglatinsquaresofsize10.theresultsareshownintable1.not workisnecessarytodiscoverhowtopredicttheseratios. test.aswellweincludeagraphv900whichisa900vertexgraphgenerated graphsofvarioussize,generatedinasimilarfashionasthoseintheprevious nautyismorelikelytondandexploitoneofthose\breaks". Onenaltestweperformedwastotryandmeasuretheamountofincrease, ndegreenoswaps(sec)oneswap(sec)blowupratio Table1:Theeectofgraphparametersettingsontheblowupratio Conclusion ThetheoryofNP-completenesshasallowedresearcherstoconcludethatmany timesolution,ortobecontentwithalgorithmsthatmayhaveverybadworstcasebehaviour.asthelargeamountofresearchonthecomplexityofgraph seemtoguidetheresearcher.oneisunsureofwhethertosearchforapolynomial problemsareeitherpolynomiallysolvable,orareprobablyintractable.graph isomorphismshows,researchersaretryingtoremedythatproblem,believing isomorphismisoneofthefewproblemsforwhichnp-completenessdoesnot tobecompletelydetermined.insteadalargebodyofworkhasconcentrated generalproblemveryfastonaverage.whenthegraphsbeingaveragedover tionsthatthecomputingcommunityisnotcontenttowaitforitscomplexity onsolvingrestrictedversionsofgraphisomorphismveryquickly,orsolvingthe largerrealmofcomplexityanalysis. thatadeeperunderstandingofgraphisomorphismwillyieldinsightsintothe However,thegraphisomorphismproblemhassomanyinterestingapplica- 21

23 Howeverpracticalalgorithmsareveryfastwhenaveragedoverthenebulous aregeneratedinacompletelyrandomfashion,itistrivialtobefastonaverage. plausiblethattheymightariseinanapplicationwhereveryregulargraphsare generated,whichmighthaveasmallamountofnoise.onepointtonoteisthat rithmsndhard.grantedthatthesegraphswererathercontrived,butitseems thesearchspace. notionofrealworldgraphs,byusingpowerfulgrouptheorytoolstohelpprune evenfor\hard"instances,1000vertexgraphsweresolvedinlessthan10minutes.thisisquiteacontrastwithproblemslikegraphcoloring,were10minutes ona1000nodeproblemwouldnotbeconsideredalongtime,andmaybea furtherindicationofthe\almostpolynomial"natureofthegraphisomorphism problem. Asweshowed,itisnottoodiculttocreategraphswhichcurrentalgo- References [1]K.Agusa,S.Fujita,M.Yamashita,andT.AE.Onneuralnetworksfor graphisomorphismproblem.inthernns/ieeeconferenceonneuroinformaticsandneurocomputers,pages1142{1148,rostov-on-don,russia, [4]L.Babel.Isomorphismofchordal(6,3)graphs.Computing,54:303{306, [3]L.Babai.Moderatelyexponentialboundforgraphisomorphism.Proceed- [2]A.Aho,J.Hopcroft,andJ.Ullman.TheDesignandAnalysisofComputer ingscience117,pages34{50,1981. ingsofthefundamentalsofcomputingscience,lecturenotesincomput- Algorithms.Addison-Wesley,Reading,1974. October1992. [6]L.BabiaandL.Kucera.Canonicallabelingofgraphsinlinearaveragetime. [7]D.Basin.Atermequalityproblemequivalenttographisomorphism.InformationProcessingLetters,51:61{66,1994. [5]L.Babia,P.Erdos,andS.Selkows.Randomgraphisomorphism.SIAM InProceedingsofthe20thIEEESymposiumonFoundationsofComputing Science,pages39{46,1979. JournalofComputing,9(3):628{635,August [9]K.Booth.Problemspolynomiallyequivbalenttographisomorphism.TechnicalReportCS-77-04,UniverisityofWaterloo,1979maticindexonpartialk-trees.JournalofAlgorithms,11:631{643,1990. [8]H.Bodlaender.Polynomialalgorithmsforgraphisomorphismandchro- 22

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