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1 GreedyApproximationsofIndependentSetsinLowDegree Magn sm.halld rsson GraphsKiyohitoYoshiharay lemincubicgraphsandgraphsofmaximumdegreethree.thesealgorithmiterativelyselect verticesofminimumdegree,butdierinthesecondaryruleforchoosingamongmanycandidates.westudythreesuchalgorithms,andprovetightperformanceratios,withthebest withthealgorithmwiththebestperformanceratioknownof1:2. algorithmthatgreedilyselectsverticesofminimumhasaperformanceratioatleast1:25on degree-threegraphs,evenifgivenanoracletochooseamongcandidateverticesofminimum degree. Wealsoshowcertaininherentlimitationsinthepowerofthisfamilyofalgorithm:any Weinvestigatethepowerofafamilyofgreedyalgorithmsfortheindependentsetprob- Abstract onebeing9=71:28.allofthesealgorithmsarepracticalandruninlineartime,incontrast AnindependentsetofagraphGisasubsetofverticesinwhichnotwoareadjacent.TheMax 1IndependentSetproblemsisthatofndinganindependentsetofmaximumcardinality.Itis Introduction measuredbytheperformanceratio,orthemaximumratioofthesizeofanoptimalsolution(the alwaysoptimalbutareclosetooptimal.thequalityofanapproximationalgorithmisgenerally toexist.itisthereforeinterestingtoexplorealgorithmsthatproducesolutionsthatarenot oneofthecorenp-hardproblems[4],andthus,polynomialtimeexactalgorithmsareunlikely sizeofthemaximumindependentset)tothesizeofthesolutionfoundbythealgorithm. etal.[1]showedthatitisnp-hardtoobtainaperformanceratiooflessthannforsome >0,wherenisthenumberofvertices.Onspecialclassesofgraphs,however,theproblem doesadmitconstantfactorapproximations. Theindependentsetproblemisknowntobehardtoapproximateongeneralgraphs.Arora followingapaperofhochbaum[9],therehasbeenaurryofresultsontheapproximationof independentsetsinbounded-degreegraphs[3,6,7,2].thecurrentlybestratiosknownare (+3)=5formaximumdegree613[3,2],=6+O(1)forintermediatevaluesof,and Oneimportantsuchclassisthatofbounded-degreegraphs.Afteradecadeofnon-activity O(=loglog)[7,8]forlargevaluesof.. degreeisatmostthree.sincetheindependentsetproblemispolynomialsolvablewhenmaximum degreeistwo,thisproblemcanbethoughtofastheinitialfrontierofnp-hardnessofthe problem.also,manyoftheresultsforhigherdegreesusereductionstolowerdegreecases,in Inthispaperwefocusonacentralcaseofbounded-degreegraphs,namelywhenthemaximum atjapanadvancedinstituteofscienceandtechnologyhokuriku,ibmtokyoresearchlab,andmaxplanck InstitutfuerInformatik. Contactauthor.ScienceInstitute,UniversityofIceland,IS-107Reykjavik,Iceland.Researchpartlyperformed ydepartmentofcomputerscience,tokyoinstituteoftechnology 1

2 three.theproblemremainsnp-hardandmaxsnp-hard(hardtoapproximatewithinsome casetranslatetoimprovementsforallodddegrees. whichthedegree-threeplaystheroleofthebasiscase[3,7,5,2],andimprovementsforthat xedconstantgreaterthanone)evenunderthesestrongrestrictions. Weadditionallyconsidercubicgraphs,i.e.3-regulargraphs,whereallverticesareofdegree thatattainsaperformanceratioof1:25.thishasrecentlybeenbroughtdownto1:2byberman graphs.hochbaum[9]presentedanalgorithmwitha1:5ratio,thatrunsintimeproportional tobipartitematchingoro(n1:5).bermanandf rer[3]gaveapowerfullocalsearchapproach Letusreviewtheknownresultsaboutapproximatingindependentsetsindegree-three andfujito[2]usingadditionaltricks.thedisadvantageofthisapproachisaphenomenallyhigh analysis[7]thecomplexityappearstobenolessthann50.inresponsetothis,halld rssonand timecomplexity:theanalysisof[3,2]yieldsaboundofatleastn2100,whileevenwithatighter Radhakrishnan[7]gaveascaled-downversionofthelocalsearchapproachof[3]whichrunsin lineartimewithaperformanceratioof1:4.generalizations[5]leadtoa1:33+ratiointime O(exp(1=)n). andrepeatuntilthegraphisempty.thisapproachisnon-deterministicinthechoiceofa vertexofminimumdegree,addthevertextothesolution,removethevertexanditsneighbors, particularvertexofminimumdegree.thebasicalgorithm,selectinganarbitraryminimum Thealgorithmparadigmthatweconsiderinthispaperisthatofgreedyalgorithms:selecta alwayschooseminimum-degreevertices,butwithdierentrulestodecideamongcandidate performanceratioondegree-threegraphswasshowntobe5=3. degreevertex,wasanalyzedindetailbyhalld rssonandradhakrishnan[6].inparticular,the vertices.typically,thealgorithmsattempttoeliminatemorethantheminimumnumberof edgesineachreduction,orpreferreductionsthatcomparewellwiththeoptimalsolution.in Weconsiderheregreedyalgorithmswithmoregoal-directedselectionrules.Thealgorithms summary,ourresultsareasfollows: 1.Thebasicgreedyalgorithmattainsaperformanceratioof3=2oncubicgraphs.Further 2.Amodiedgreedyalgorithmispresentedthatattainsaratioof3=2on(general)degreethreegraphs.Theratioimprovestoaratiothatapproaches4=3oncubicgraphswithhigh restrictingtheinputtographsofhighoddgirthyieldsnofurtherimprovements. 3.Asecondmodiedgreedyalgorithmispresentedthatattainsaratioof9=71:28on oddgirth. 4.Anygreedyalgorithmisshowntohaveaperformanceratioatleast1:25ondegree-three degree-threegraphs. Thepaperisorganizedintosectionsfollowingtheabovelist. algorithm. graphs.thus,thewholefamilyhaslimitationswhicharenearlymatchedbyoursecond 22.1Notation Weusestandardsymbolsandnotations.TheinputgraphG=(V;E)isassumedtobeof Preliminaries thenumberofvertices,mthenumberofedges,theindependencenumber(i.e.sizeofthe maximumdegreethree,withfurtherrestrictionsexplicitlystatedwheninplace.letndenote 2

3 optimalindependentset).foravertexv,n(v)denotestheneighborhoodofv,orthesetof adjacentvertices. Adef performanceratioofaisdenedasthemaximumapproximationratiooverallinputgraphs,or denotedbya(g),andtheapproximationratioa(g)isdenedasa(g)def ForanalgorithmAforMaxIndependentSet,thesizeofthesolutionproducedonGis WeletIdenoteaxedbutarbitrarymaximumindependentsetinG.LetOutdenotethe =maxga(g).weareprimarilyinterestedinthelimitofthisvalueasngoestoinnity. =(G)=A(G).The numberofedgeswithbothendpointsinv?i. itisessentialtohaveatonesdisposalagoodupperboundoftheoptimalsolution.thenumber Inordertoanalyzetherelativevalueofaheuristicsolutioncomparedwithanoptimalsolution, OutofedgesoutsidesomemaximumcardinalitysolutionIplaysacrucialrole. 2.2Upperboundingtheoptimalsolution Lemma2.1Foradegree-threegraphG, (G)n?m=3?Out=3: (1) ofendpointsinv?iisatmost3(n?jij).thus, Proof.EachedgehaseitheroneendpointinIorbothendpointsinV?I.Thetotalnumber which,whenrearranged,yieldstheclaim. Inacubicgraph,m=3n=2,andtheinequalitybecomesn=2?Out=3. m3(n?jij)?out; 3Werstconsiderthewell-knownGreedyalgorithm,whichwelabelhereasGreedy.Thealgorithm proceedsalongasequenceofiterationsorreductions,eachofwhichconsistsofthefollowingtwo GreedyAlgorithmonCubicGraphs steps:somevertexofminimumdegreeisaddedtothesolution,andthevertex,itsneighbors,and havebeendeletedfromthegraph.sinceneighborsofaselectedvertexareimmediatelydeleted, allincidentedgesareremovedfromthegraph.thealgorithmterminateswhentheallvertices thesolutionconsistingoftheselectedverticesformaproperindependentset.bymaintaining trackofthedegreesofthevertices,thealgorithmcanbeimplementedino(n+m)time. ondegree-threegraphswasshowntobe5=31:66.abetterratioispossibleinthecaseof cubicgraphs. Greedywasanalyzedforbounded-degreegraphsin[6],whereitsapproximationperformance nected.observethatgreedypicksavertexofmaximumdegreeatmostonce,sincenoproper Theorem3.1TheperformanceratioofGreedyoncubicgraphsis3=2. step.thatis, inducedsubgraphcanberegular.thus,atmostthreeverticesaredeletedinallbuttherst Werstarguetheupperbound.Assumewithoutlossofgeneralitythatthegraphiscon- Min0(n?1)=3: 3

4 If1or2,thenGreedyn=3;if3or4,thenn=2?1=3byLemma2.1.Ineithercase,the Remark.ItcanbearguedthatGreedyndsanoptimalsolutioninregularbipartitegraphs. performanceratioisatmost3=2. Considerthelastreductionmade.Thedeletedverticesmustformacliqueon1to4vertices. 3=2?3=(2n?2). ByapplyingLemma2.1,thatimpliesaperformanceratioofatmost(n=2?1=3)=((n?1)=3)= graphisconstructedfromthreeunits:frontunit,backunit,andmultiplecopiesofrepetition achainwiththreeedgesbetweenadjacentcopies.thechainisankedontheendsbythefront units.therepetitionunitsareintheformofa12-cyclewiththreecords,andareconnectedin WenowconstructahardgraphforGreedythatshowsthattheaboveratioistight.The andrearunits,bothintheformofacompletebipartitegraphk2;3,withthethreeverticesinone bypicture,infigure1. partitionconnectedtotheendsoftherepetitionunitchain.thegraph,g0,isbestdescribed 2 4 uf vf breakingchoices.thealgorithmstartsbychoosingvfandufofthefrontunit.ontherst Figure1:AhardgraphforGreedy. front unit 1 3 repetitionunititchoosesvertices1through4inthatorder.thisleavesanidenticalgraphless Weindicatetheworst-casebehaviorofGreedybypresentingaparticularsequenceofsymmetry- back unit asinglerepetitionunit.hence,thealgorithmpicksthefourshadedverticesofeachrepetition unit,endingwiththreeverticesfromtherearunit. solutioncontains6`+4.thetotalnumberofverticesis12`+10.hence,theapproximation ratioofgreedyong0is: If`isthenumberofrepetitionunits,thealgorithmnds4`+5verticeswhiletheoptimal 3=2.Observethatthesameholdsevenifthegraphsarerequiredtobetriangle-free. WeconcludethattheperformanceratioofGreedyoncubicgraphsasymptoticallyequals 0(G0)=6`+4 4`+5=32? 2(n+5): 21 4TheworstcasebehaviorofGreedy,asseenwhenappliedtothehardgraphsG0intheprevious section,suggestsadirectionformodifyingthestrategyofthealgorithm.asituationwhere AModiedGreedyAlgorithm Greedyappearstobeweakiswhentherearemanyverticesofminimumdegree.Inthissection, weproposeamodiedversionofgreedy,namedmoreedges,whichconsidersthedegreesof verticesadjacenttoavertexasacriteriaforselectingthevertices.thecriteriais: Whenminimumdegreeistwo,selectwheneverpossibleavertexwithaneighbor ofdegreethree. 4

5 solution. composedofseveraldisjointcycles.onthatremainingportion,moreedgesobtainsanoptimal Ifnosuchvertexexistsandtheminimumdegreeistwo,thenwecanshowthatthegraphis degree-threegraphs. Theorem4.1TheperformanceratioofMoreEdgesondegree-threegraphsis3=2. Wendthatthismodiedalgorithmyieldsanimprovementoverthe5=3ratioofGreedyon Upperbound Theoperationofthealgorithmcanbebrokenupintoreductions,eachofwhichconsistsof theadditionofasinglevertextothecurrentsolutionandthedeletionofthisandneighboring verticesalongwiththeincidentedges.iftherecursivedescriptionofthealgorithmismade iterative,areductioncorrespondstoasingleiteration.an(i;j)-reductionreferstoonewhere i?1verticesandjedgesaredeleted. 0,0 1,2 1,1 1,3 2,3 2,4 Selected Deleted Incident 2,4 2,5 2,5 2,6 3,6 3,7 3,8 3,9 theirneighbors(whicharealsodeleted)areingrey,andotherincidentverticesareinwhite. TheformofthepossiblereductionsaregiveninFigure2.Theselectedverticesareinblack, Figure2:Theformsofthevariousreductions. nalityindependentsetiforcomparison. (2;4)and(2;5)reductionsappearintwodierentguises. n(r) Weconsiderthefollowingmeasuresofeachreductionr.Here,wexsomemaximumcardi- Out(r)NumberofdeletededgeswithbothendpointsinV?I. e(r) (r) Numberofverticesdeleted Ourprimarycostmeasureofeachreductionrisgivenby: Numberofedgesdeleted NumberofthedeletedverticesthatbelongtoI forout(r),e(r)arelowerbounds,while(r)andf(r)areupperbounds. Table1givesconservativeboundsforthesemeasuresoneachtypeofreduction.Thevalues f(r)=3n(r)?e(r)?out(r)+(r): 5

6 r(0;0) 1n(r)e(r)Out(r)(r)f(r) (2;4) (2;3) (1;x) y (3;8) (3;7) (3;6) (2;5+) Table1:BoundsonmeasuresofthereductionsperformedbyMoreEdges (3;9) whichvertexitstartswith.thus,itcausesnoharmifweassumethatitchoosesavertexfor of(2;4).whena(2;4)-reductionoccurs,thegraphnecessarilyconsistsofdisjointcycles.the algorithmwilladdthesamenumberofverticestothesolutionfromagivencycle,nomatter ThevaluesinTable1areeasilyveriedfromFigure2,withtheexceptionofthevalue whichatmostoneneighborbelongstoagivenmaximumindependentset. Theclaimoftheupperboundnowfollowseasilyfromthef-valuesofTable1,andLemma2.1: Lowerbound 6t=Xr6Xrfr=3n?e?Out+4: WeconstructahardgraphforMoreEdgesasinFigure3.Itisachainofsimpleunitswith sixverticeseach.eachunitformsasix-cyclewithonecordbetweenthethirdandthefth vertex.thelastvertexineachunitisalsoadjacenttotherstvertexinthesubsequentunit. Formally,weconstructafamilyofgraphsGq,withverticesverticesvi;1;:::;vi;6andedges (vi;j;vi;j+1);(vi;3;vi;5);(vi0;6;vi0+1;1),wherei=1;:::q,j=1;:::6andi0=1;:::q?1. optimalsolutionwillcontainthesecond,fourthandsixth.onlyonthelastunitwillmoreedges MoreEdgesmaybeassumedtoselecttherstandthethirdvertexofeachunit,whilethe Figure3:InitialportionofahardgraphforMoreEdges. 3=2?(1=n). alsondthreevertices.hence,theperformanceratioofmoreedgesisnobetterthan3q=(2q+1)= Furtherresults Wehavefurtheranalyzedthealgorithmforclassesofcubicgraphs.Inparticular,thealgorithm attainsperformanceratiosof17=121:42oncubicgraphs,29=211:38oncubictriangle-free tightastherearegraphswheretheseratiosoccur.weomitthedescriptionsforreasonsofspace. graphs,andingeneral4=3+1=(9k+3)oncubicgraphsofoddgirth2k+1.thosevaluesare 6

7 algorithmperformsthefollowingtwotypesoftransformationswheneverpossible. Weconsiderinthissectionastillstrongermemberofthegreedyparadigm.Inparticular,the 5 ASecondImprovedAlgorithm BranchyreductionWhentwoverticesvanduofdegreetwoareadjacent,someoptimal solutionwillcontainexactlyoneofthesevertices.wecantransformthegraphintoa graphg0thatcontainsallverticesbutvanduandhastheotherneighborsofvandu adjacent.toensurethatmulti-edgesdonotappear,weinsistthatnothirdvertexbe adjacenttobothvandu.thesolutionoftheheuristicwillcontaintheheuristicsolution SimplicialreductionAsimplicialvertexisonewhoseneighborhoodformsaclique.Anoptimalsolutioncancontainatmostonevertexfromthisopenneighborhood,henceselecting eectisoptimal. ong0alongwithoneofvandu.thisisacaseofadelayed-commitmentreduction,whose Simplicialreductionsappearas(1;1?3),(2;3?5)and(3;6)-reductions.Branchyreductions onewhosetwoneighborsareadjacent,wheneverpossible. asimplicialvertexisalwaysoptimal.inparticular,whenminimumdegreeistwo,weselect appearas(1;2)-reductions.thesetrickshaveearlierbeenusedin[2]. Simplicial(G) repeat performreductionsinthefollowingorderofpreference: 1.branchy,simplicial,(2;6) end untildone 3.(3;9) 2.(3;8) algorithmcanbeimplementedinlineartime. Bykeepingtrackoftheshapeoftheneighborhoodofeachvertexinthecurrentgraph,the 4=3+1=[5].Further,byusingthisalgorithmasthesubroutinefordegreethreegraphsin previousbestratioclaimedforanalgorithmwithlow-polynomialtimecomplexity[6,7,5]was theschemaof[7](originatingin[3]),weobtainsimilarimprovementsfortheindependentset Thisalgorithmattainsaratioof9=71:28ondegree-threegraphs.Incomparison,the Theorem5.1TheperformanceratioofSimplicialongraphsofmaximumdegreethreeisexactly probleminotherclassesofbounded-degreegraphs. 9=7ẆerstgiveasimpleconstructionforthelowerboundinFigure4.Firstbuildaunitwith Wedevotetherestofthissectionforprovingourmainresult. byaddingonevertexuconnectedtothev1'softhreeunits. andfthandseventh.allverticesareofdegreethree,exceptv1.thehardgraphisobtained sevenvertices,v1;:::v7formingacyclewithchordsbetweensecondandfourth,thirdandsixth, maychooseanyvertexbutuandv1vertices.onetie-breakingchoiceistoselectv2,followedby v7,u,andtwoverticesfromeachofthetworemainingunits,foratotalofseven.ontheother hand,theoptimalsolutionconsistsoftherst,thirdandfthvertexofeachunit,foratotalof Fortheinitialchoice,thealgorithmwillprefera(3;8)-reductionovera(3;9)-reduction,and nine. 7

8 v 4 v 5 Figure4:AhardgraphforSimplicial. v v 4 5 v 4 v 5 v 3 v 6 v 3 v 6 v 3 v 6 v 2 v 7 v 2 v 7 v 2 v v 1 v1 v isgivenby: 5.1Upperbound UsingthesamemeasuresofthereductionsasforMoreEdges,ourcostmeasureforthisalgorithm 7 1 u problembyconsideringshortsequencesofreductions,oridiomsaswecallthem,andshowing thatthecostmeasureonthesecombinationsbehaveasdesired. Unfortunately,thismeasureistoolargeon(1;1)and(2;3)reductions.Wealleviatethis g(r)=6n(r)?2e(r)?2out(r)+(r): stringcanbelexicallypartitionedintostringsfromarestrictedclass. Claim5.2Thefollowingisanalphabetforthereductionsequenceofthealgorithm: Letusviewtheexecutionofthealgorithmasastringofreductions.Wearguethatthatthis [f(2;6);(3;8);(3;9)gf(1;2);(2;4)g(1;1)],[f(1;3);(2;5);(2;6);(3;9)gf(1;2);(2;4)g(2;3)], f(0;0),(1;2),(1;3),(2;4),(2;5),(2;6),(3;8),(3;9), ThefollowingobservationshavebearingonClaim5.2. [(3;8)f(1;2);(2;4)g(2;3);f(2;5);(2;6)g]. 2.Thepossibilityoftheidiom[(3;9);(2;3);(2;3)]iseliminatedbythethirdcaseofthe 1.A(3;7)-reductionisimpossible,sincesometwooftheverticesinthereductionwouldgive algorithm. risetoa(3;8)-reduction. 4.Followingthesequence[(3;8);(2;3)],theremaininggraphwillbecubicexceptfora 3.Theonlyreductionsthatcanprecedea(2;3)reductionare:(1;3),(2;5),(2;6),or(3;9). Thisignores(1;2)and(2;4)-reductionswhichmaybeinterspersedinvariousways. singlevertex.henceonly(2;5)or(2;6)reductionscanimmediatelyfollow.further, measureofeachreductionintheidiom.further,foranidiom,lett()denotethenumberof Wegeneralizethemeasuresofreductionstomeasuresofidioms,bytakingthesumofthe [(3;8);(2;3);(2;5);(2;3)]isnotpossible. fundamentalidiomsinthealphabet.fortheidiomsthatinclude(2;3),wehaveomittedthe interspersed(1;2)and(2;4)reductions,andcountedthemasindividualidioms.theidioms reductionswithin. involving(1;1)havealsobeencompacted. Table2giveslistslowerboundsforOut()andupperboundsfor()andg()forthe oftheindividualreductionsintheidiom. ThefollowingvaluesaredierentfromTable1oraredierentfromthesumsofthevalues 8

9 (0;0) (1;1) (1;2) 1n()e()Out()()g()g()=t() (1;3) 012 (2;4) 2 3 (2;5) 4 (2;6) (3;8) f(2;6);(2;3)g f(1;3);(2;3)g f(3;9);(2;3)g :5 f(2;5);(2;3)g f(3;8);(2;3);(2;5)g10 f(3;8);(2;3);(2;6)g : andOutfor(2;4)and(2;5):Noticethatonlythelatterformofthesereductionsin Figure2cannowappearduetothepreferencetothedelayed-commitmentreduction. Table2:BoundsonmeasuresofthereductionsperformedbySimplicial 2.Outfor(1;1):Whicheveridioma(1;1)-reductionappearsin,anadditionaledgemustbe 3.TheOutvaluesof(2;5)(2;3)and(3;9)(2;3)andvaluesof(2;6)(2;3)and(3;9)(2;3): Thereasonsareclearwhenwelookatthesubgraphsinducedbythesepairsofreductions. outsideofi. theindependencenumberoroutsideedgesinthesubgraphinducedbytheidiom. vertices,thus,havinginterspersedwithinanidiomdoesnotaectanyargumentabout (1;2)and(2;4)-reductionsyieldoptimalresultsonthesubgraphinducedbythedeleted alwaysatmost9.9t=9xt()xg()6n?2e?2out+7: ThetheoremnowfollowsfromLemma2.1alongwiththefactthatg()=t()inTable2is Greedy,MoreEdgesandSimplicial.Thesetwoalgorithmshaveacommonbasicstrategyofremovingavertexwiththeminimumdegreeinacurrentgraphateachstage.Wecaneasilysee bytheirhardgraphsthattheweaknessesofthegreedyalgorithmsappearwhentheyhaveseveralwaysofchoosingaminimum-degreevertex.ifwecouldgivethesealgorithmssomeadvice suchthattheycouldproceedoptimallywhenevertheyfaceabranchroad,howmuchwouldthe algorithmsimprove?orwouldanalgorithmthatwasgivenperfectadvicenecessarilyndan optimalsolution? 6Intheprevioussections,weconsideredtheperformanceratiosofthreegreedyalgorithms: AlgorithmwithAdvice,Ultimate 9

10 oracleforselectingamongalternatives.theonlyrequirementisthatthealgorithmmustchoose oneoftheminimumdegreeverticesatanystep.werefertothisultimatealgorithmasultimate, indicatingthatthealgorithmhasinnitevisibility(orarbitrarydistancefromthegivennode) Inthissection,westudythepowerofalgorithmsthataregiventheadditionalbenetofan forchoosingamongminimumdegreevertices. cannotndanoptimalsolution.infact,itcannotguaranteeamuchbetterperformanceratio thanthealgorithmoftheprevioussection.thisrevealsalimitationonthepowerofthefamily ofgreedyalgorithms. WeshallshowthatevenbyemployingUltimate,thereremainsgraphsforwhichUltimate setofh4issix,whereasultimatendsonlyvevertices.oneoptimalsolutionconsistsofthe WerstformasubgraphH4asontheleftofFigure5.Thesizeofthemaximumindependent Figure5:ConstructionofH4,H6,andG1. shadedverticesinfigure5.weconstructapseudobinarytreeh6withfourh4'sasleaves.it byrepeatingthesameoperationq?2times. isillustratedinthecenteroffigure5.weformapseudobinarytreeh2qwith2q(q2)levels HEU(H2q)=1+4HEU(H2(q?1))= q?2+4q?15=4q?116=3?1=3; Thesizeofthesolutionfoundbythealgorithmis: whilethesizeoftheoptimalsolutionis: Asqgrows,theratioofHEUtoOPTapproaches5=4. OPT(H2q)=2+4OPT(H2(q?1))=2( q?2)+4q?16=4q?120=3?2=3: rightoffigure5inordertomaketheentiregraphcubic,andcalltheresultinggraphg1. UltimatepicksanyvertexinaH2qattherststep.ItiseasytoverifythatMin1thenproceeds Wecanalsoobtainahardnessresultforregular(i.e.cubic)graphs.JointwoH2qsasonthe optimallyonthathalfofthegraph.thatleavestheotherh2qleft,forwhichthealgorithmwill, byinduction,benon-optimal. AnoptimalsolutionofG1containsallverticesontheevenlevelsofeachH2q.Supposethat Thus, highoddgirth),byreplacingthetrianglesatthebottomofh5byave-cycle(oranappropriately Wecanobtainsimilarhardnessresultsfortriangle-freegraphs(ormoregenerallygraphsof 1(G1)= OPT(H2q)+HEU(H2q)= OPT(H2q) 20+16=10 9=1:1: largeoddcycle)andconnectingthepairstogetherasneeded. triangle-freecubicgraphs. Theorem6.1Anygreedyalgorithmthatselectsverticesofminimumdegreemusthaveperformanceratiosatleast:1:25,fordegree-three;1:11,forcubicgraphs;and16=151:06,for 10 H 4 H6 H4 H4 H4 H4 H2q root root H2q

11 Acknowledgments forinformativecommentsanddiscussions. References WearemuchindebtedtoProfessorOsamuWatanabeandProfessorJaikumarRadhakrishnan [1]S.Arora,C.Lund,R.Motwani,M.Sudan,andM.Szegedy.Proofvericationandhardness [2]P.BermanandT.Fujito.Ontheapproximationpropertiesofindependentsetproblemin ofapproximationproblems.focs1992. [3]P.BermanandM.F rer.approximatingmaximumindependentsetinboundeddegree degree3graphs.wads1995. [4]M.R.GareyandD.S.Johnson.ComputersandIntractibility:AGuidetotheTheoryof graphs.soda1994. [5]M.M.Halld rsson.approximatingdiscretecollectionsvialocalimprovements.soda1995. [6]M.M.Halld rssonandj.radhakrishnan.greedisgood:approximatingindependentsets NP-completeness.Freeman,1979. [7]M.M.Halld rssonandj.radhakrishnan.improvedapproximationsofindependentsetsin bounded-degreegraphs.swat1994. insparseandbounded-degreegraphs.stoc1994.toappearinalgorithmica. [9]D.S.Hochbaum.Ecientboundsforthestableset,vertexcover,andsetpackingproblems. [8]M.M.Halld rssonandj.radhakrishnan.improvedapproximationsofindependentsetsin Disc.AppliedMath.,6:243254,1983. bounded-degreeviasubgraphremoval.nordicj.computing,1(4):475492,