problemofndinganindependentsetofmaximumcardinalityisoneofthefundamentalcombinatorialproblems.itisknowntobenp-complete,evenforbounded-degreegraphs,and

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1 Greedisgood:Approximatingindependentsetsinsparseand Magn sm.halld rssonz bounded-degreegraphsy methodforndingindependentsetsingraphs.weshowthatitachievesaperformanceratio Theminimum-degreegreedyalgorithm,orGreedyforshort,isasimpleandwellstudied Abstract JaikumarRadhakrishnanx ecientparallelanddistributedalgorithmattainingtheperformanceguaranteesofgreedy. algorithmasafunctionoftheindependencenumber,aswellasageneralizationoftur n's analysisyieldsaprecisecharacterizationofthesizeoftheindependentsetsfoundbythe bound.wealsoanalyzethealgorithmwhenrunincombinationwithaknownpreprocessing technique,andobtainanimproved(2d+3)=5performanceratioongraphswithaverage degreed,improvingonthepreviousbest(d+1)=2ofhochbaum.finally,wepresentan of(+2)=3forapproximatingindependentsetsingraphswithdegreeboundedby.the problemofndinganindependentsetofmaximumcardinalityisoneofthefundamentalcombinatorialproblems.itisknowntobenp-complete,evenforbounded-degreegraphs,and 1Introduction Anindependentsetinagraphisacollectionofverticesthataremutuallynon-adjacent.The selectsavertexofminimumdegree,deletesthatvertexandallofitsneighborsfromthegraph, withagoodperformanceratio,whichisaboundonthemaximumratiobetweentheoptimal solutionsize(i.e.theindependencenumber)andthesizeofthesolutionfoundbytheheuristic. andrepeatsthisprocessuntilthegraphbecomesempty.asadelightfullysimpleandecient thereforenoecientalgorithmsareinsight. theindependentsetprobleminbounded-degreegraphs.inparticular,weseekanalgorithm algorithm,thegreedymethoddeservesaparticularlydetailedanalysis.itisalreadyknownto Oneofthemostubiquitousheuristicmethodsforthisproblemisthegreedyalgorithmwhich Giventhehardnessofexactcomputation,weareinterestedinapproximationalgorithmsfor andndingoptimalindependentsetsintrees,series-parallel,cographs,andgraphsofdegreeat most2. extentofitsperformancehasapparentlynotbeendeterminedbefore.thebestratiopreviously possessseveralimportantqualities:attainingthetur nbound,anditsgeneralizationinterms ofdegreesequences[30,8];almostalwaysobtainingasolutionatleasthalfthesizeofanoptimal solutioninarandomgraph[23];yieldinganon-trivialgraphcoloringapproximation[16],anda lightcoloringwithasmallchromaticsum[19],whenappliediterativelyasacoloringmethod; JapanAdvancedInstituteofScienceandTechnologyHokuriku. GordonGekko[29]. yapreliminaryversionofthispaperappearedinthe26thacmsymposiumontheoryofcomputing,1994. xtheoreticalcomputersciencegroup,tatainstituteoffundamentalresearch,bombay,india WhiletheperformanceratioofGreedyhasbeenanalyzedbeforetosomeextent,thetrue 1

2 averagedegreed[14]. claimedforgreedywas 1ongraphswithmaximumdegree[27]andd+1ongraphsof numberandthenumberofvertices. oftur n'sboundthatincorporatestheactualindependencenumberofthegraph,andgivea ittoimprovethebestperformanceratioknownintermsofaveragedegreeto(2d+3)=5,but general,tightexpressionofthesizeofthesolutionfoundasafunctionoftheindependence boundof(d+2)=2intermsofaveragedegree.intheprocess,wegiveanaturalextension performanceratioof(+2)=3intermsofmaximumdegree,andanasymptoticallyoptimal WefurtheranalyzeGreedyextendedwithapreprocessingmethodofHochbaum[14].Weuse OurmainresultisthatGreedyismuchbetterthanpreviouslyclaimed.Weobtainatight naturallyleadstoaparallelanddistributedalgorithminheritingtheapproximativeproperties algorithmandsomeofitspropertiesincludingthetur nbound,andreviewotherresultson theperformanceguaranteesclaimedabove;infact,itholdsforanyvertexwhosedegreeisat mosttheaverageofthedegreesofitanditsneighbors.thisisalocallyevaluatedpropertythat ofgreedy,fortherstnon-trivialsuchapproximationsknowntous. showittobeoflimiteduseintermsofmaximumdegree. approximatingthisproblem.weanalyzegreedyindetailinsection3,startingwithageneralizationofthetur nboundinsection3.1,followedbytightperformanceratiosinsection3.2, Theremainderofthepaperisorganizedasfollows.InSection2wepresenttheGreedy ItfollowsfromouranalysisthatgloballyminimumdegreeisnotrequiredforGreedytoachieve andlimitationsonitsperformanceinsection3.3.insection4,weconsiderimprovementsobtainedbyadditionallyapplyingapreprocessingtechnique,anddescribeinsection5aparallel denotedg=(v;e),ndenotesthenumberofvertices,themaximumdegree,dtheaverage algorithmattainingtheboundsprovedforgreedy. 1.1Notation Weusefairlystandardgraphnotationandterminology.Forthegraphinquestion,usually degree,theindependencenumber(thesizeofthelargestindependentset),andtheindependencefraction(thatis,=n).foravertexv,d(v)denotesthedegreeofv,andn(v)theset WefocusonGreedy,denotedbyGr. G=(V;E).TheperformanceratioAofAisdenedby ofneighborsofv. ForanindependentsetalgorithmA,A(G)isthesizeofthesolutionobtainedbyAongraph 2TheGreedyAlgorithmandRelatedResults A=A(n)=max G;jGj=n(G) 2.1TheGreedyAlgorithm A(G): pendentsetbyselectingsomevertexofminimumdegree,removingitanditsneighborsfrom thegraph,anditeratingontheremaininggraphuntilempty. Theminimum-degreegreedyalgorithm,orGreedyforshort,incrementallyconstructsaninde- Greedy(G) I whileg6=;do ; 2

3 end odg OutputI Choosevsuchthatd(v)=min I I[fvg G fvg[n(v) w2v(g)d(w) independentofthedegree. Thatis,viscriticalifitsatises:d2(v)X Thealgorithmcanbeimplementedintimelinearinthenumberofedgesandvertices, Avertexofminimumdegreeiscritical,hencesuchanodealwaysexists.Althoughwestate Wecallanodecriticalifitsdegreeisatmosttheaverageofthedegreesofitanditsneighbors. selectedvertexiscritical. toaniteration.inareduction,avertexisselected,addedtothesolution,andthenremoved Greedywiththisminimum-degreepivotingrule,theonlypropertythatweshalluseisthatthe Consideranexecutionofthealgorithmtobeasequenceofreductions,eachcorresponding w2n(v)d(w): (1) Thisallowsustoboundfrombelowthenumberofedgesremovedineachstep. alongwithitsneighborhoodfromthegraph.lettdenotethenumberofreductionsanddithe ofthedegreesofthedi+1verticesremovedinthei-threductionmustbeatleastdi(di+1). degreeintheremaininggraphofthei-thvertexselected,i=1;2;:::t.thenumberofvertices removedinthei-threductionisthusdi+1. Aclassicaltheoremingraphtheory,duetoTur n[28],statesthatforanygraphg, 2.2PreviousresultsonGreedy Themainpropertyofthealgorithmthatweshallbeusinginouranalysisisthatthesum whichheshowedwasattainedbygreedy.thisresultactuallyfollowsfromanearliertheorem equalaspossible.wei[30](see[10])provedanextension, andthattheinequalityistightonlyforthegraphconsistingof(g)disjointcliquesofsizeas (G)Xv2V1=(d(v)+1); (G)n=(d+1); Theorem1Grn ourcentralresultsbuilddirectlyuponit. originalgraph.theproofofhistheoremcanalsobeseentoindirectlyrefertogreedy. oferd s[8](see[10]),whichstatesthatifagraphcontainsnoindependentsetofsizek+1, thenthereisgraphconsistingofkdisjointcliqueswhosedegreesequencedominatesthatofthe fromthegraph.wecountthenumberofverticesandedgesdeletedineachreduction. performsasequenceoftreductions,eachtimepickingavertexanddeletingitanditsneighbors Proof.TheproofisavariationoftheonegivenbyHochbaum[14].TheGreedyalgorithm WeincludeaproofhereofthefactthatGreedyattainstheTur nbound,sincetheproofsof d+1 3

4 stepiisatleastdi(di+1),andthusthenumberofedgesdeletedisatleasthalfthatamount. Summingoverallthereductions:d2n=jEjtXi=1 SinceGreedyalwaysselectsacriticalvertex,thesumofthedegreesoftheverticesdeletedin Theremovalofverticesineachreductionpartitionsthevertexset;thus: txi=1(di+1)=n di+1 2!: (2) Wenowadd(2)andtwice(3),andobtain: (d+1)ntxi=1(di+1)2: (3) vertexofmaximumdegree,untilnoedgeremains.surprisingly,thisalgorithmalsoattainsthe UsingtheCauchy-Schwarzinequalityand(2),weget: Tur nbound,asprovedindependentlybygriggs[10]andchv talandmcdiarmid[7].its verticesfoundbygreedy. approximativepropertiesarehoweverweaker.onthegraphwith2sverticesthatiscomplete Rearrangingtheinequality,weobtainthedesiredboundont,whichispreciselythenumberof AnothersimplealgorithmoperatesbyarulethatistheinverseofGreedy:itdeletesa (d+1)nn2=t: jijedgesmustexitanindependentsetiwhileatmost(n jij)edgescanbeincidenton further.,theindependencenumberisboundedfromabovebyn=(+).thisisbecauseatleast setforanapproximationratioofs=2=(+1)=2.asaresult,wedonotconsiderthismethod canbeobtainedbyrudimentaryarguments.observethatinagraphwithminimumdegree bipartitelessasingleperfectmatching,thealgorithmmayndonlyatwovertexindependent theremainingvertices.thusinaregularconnectedcomponent,theindependencenumberisat mostn=2,foraratioof(+1)=2.thisisatmost 1for3,andwealsoknowthat Greedyisoptimalwhen=2. Anupperboundof 1oftheperformanceratioofGreedy(withtheminimum-degreerule) minimumdegreeisatleasttwo.hence,bythepreviousargument,theindependencenumberis atmost 1,soGreedyndsatleastn=independentvertices.AslongasGreedyselectsa Theaboveargumentmaybewhatisalludedtoin[27,p.306]. atmostn=(+2).combined,thisyieldsaratioof2=(+2),whichisalsoatmost 1. thetwoverticesdeletedineachstep.thus,wemayassumewithoutlossofgeneralitythatthe vertexofdegreeoneitproceedsoptimally,sincetheoptimalsolutioncancontainatmostoneof Considernowthecaseofanon-regularcomponent.Ineachstepthereisavertexofdegree verticesnotalreadyinamaximalsolutioni.4 ratioof+1.infact,aratioofholds,sinceanoptimalsolutioncancontainatmostjij Anymaximalindependentsetisofsizeatleastn=(+1),whichresultsinatrivialperformance 2.3Relatedresults

5 perboundsonthesizeoftheoptimalsolution.thetechniquewasbasedonresultsofnemhauser wend.lov sz[22]gaveanelegantproofthatcanbeturnedintoanecientalgorithm. lem.wedescribethisapproachinmoredetailinsection4.sheobtainedapproximationsof andtrotter[24]onsolutionsofthelinearprogrammingrelaxationoftheindependentsetprob- coloredwithcolorsunlessisconsistsof+1-cliqueoranoddcycle.sincewecandisposeof theexceptionsoptimally,thisyieldsastrongerboundofn=onthesizeoftheindependentset Hochbaum[14]introducedapreprocessingtechniquewhoseeectwastoobtainstrongerup- ThetheoremofBrooks[5]isanearlyresultthatstatesthatanyconnectedgraphcanbe minimumdegreeofanyinducedsubgraphofg,aswellasaratioof(d+1)=2intheunweighted weightedindependentsetsbyapplyingcoloringheuristicsandselectingtheheaviestcolorasa solution.inparticular,sheobtainedratiosof=2and(d(g)+1)=2,whered(g)isthelargest numberofverticeswhileaddingagreaternumber. approachisalocalsearchmethod,thatseeksalargersolutionprimarilybydeletingamoderate backoftheirmethodsistheexorbitanttimecomplexityofmorethanexp(3210logn).their followedbybermanandfujito[3]gavesignicantlyimprovedratiosof(+3)=5.thedraw- caseȧdecadepassedwithlittleeect.independentofthecurrentwork,bermanandf rer[4] alsoanalyzedfurtherthelocalsearchmethodoftheprevioustwopapers,decreasingthetime smalltofairlylargedegree,aswellasanasymptoticratioofo(=loglog)[11,12].wehave independentsets.thisallowsustoobtainperformanceratiosof=6(1+o(1))forgraphsof byresultsingraphtheorythatstatethatgraphswithoutsmallcliquescontainprovablylarger of[14]. deletesonewhileaddingtwoormore.theyobtainedratiosthatimprovedonthe=2bound Khannaetal.[18]alsoconsideredasimplerversionofthislocalsearchstrategy,thatmerely anecientversion,illustratingfurthertime/approximationtradeopossibilities. complexityneededfortheresultof[4],andobtainedaperformanceratioof(+3)=4[11]with Halld rssonandyoshihara[13]haveusedideasfromthecurrentpapertoanalyzemodied Alongsidewiththecurrentwork,westudiedsubgraphremovalmethodsthataremotivated theygivealinear-timealgorithmthatattainsaratioof9=71:286. versionsofthegreedyalgorithmschema.inparticular,forgraphsofmaximumdegreethree, xed)belongstotheclassofmaxsnp-completeproblemsintroducedbypapadimitriouand greatstridesinrecentyears.theindependentsetprobleminbounded-degreegraphs(foreach Yannakakis[25].Thegroundbreakingresultsonthetheoryofinteractiveproofs,resultinginthe independentsetprobleminboundeddegreegraphswithinafactorof1+isnp-hard.even landmarkpaperofaroraetal.[2],showthatthereisaxed>0suchthatapproximatingthe strongerresultsholdwhendegreerestrictionsarelifted:theindependentsetproblemcannot beapproximatedwithn,forsome,unlessp=np[2].alonetal.[1]haverecentlybeen Theprecedinghasfocusedonpossibilityresults,butnon-approximabilityresultshavemade issimilarlynp-hard. 3AnalysisofGreedy 3.1RelativesizeofGreedysolutions WestartbystrengtheningtheconstructiveversionofTur n'stheorem,byexpressingthesize abletoscalethelatterresultsdowntoshowthatapproximationonbounded-degreegraphs ourbounddominatestur n's,yieldingstrictimprovementswhenevertheindependencenumber oftheobtainableindependentsetasafunctionoftheindependencenumber.observethat exceedsthepromiseoftur n'sbound.recallthatistheindependencefraction,=n. 5

6 Theorem2Gr1+2 Proof.OurprooffollowsthatofTheorem1,whilewenowadditionallykeepcountthenumber thedi+1verticesdeletedinreductionithatarealsocontainedinthatmaximumindependent set.then: ofverticesdeletedthatbelongtosomemaximumcardinalityindependentset. Fixanindependentsetofmaximumcardinality,andletkibethenumberofverticesamong d+1+n thatnoedgecanhavebothitsendpointsinthemaximumindependentset.thus,thenumber ofedgesdeletedisatleast di+1 Recallthatthesumofthedegreesoftheverticesdeletedinstepiisatleastdi(di+1).Note 2+ ki d2n=jejtxi=1 2.Henceweobtainthefollowingstrengtheningofineq.3: txi=1ki=: di+1 2!+ ki 2!: (5) (4) Wenowadd(2),(4)andtwice(5),andapplytheCauchy-Schwarzinequalitytoobtain: cedingprooftoobtainboundsparametrizedbythemaximumdegree. Theorem3Gr1 (1 ) Rearrangingtheinequality,weobtainthedesiredboundont. Wenowturnourattentiontobounded-degreegraphs,usingtechniquessimilartothepre- (1 )+1n (d+1+)ntxi=1(di+1)2+k2i(1+2)n2=t: Proof.WeextendtheproofofTheorem2.Inthei-thstepdi+1verticesandalledgesincident onthemaredeleted.oftheseedges,letxihaveonlyoneendinthesedi+1vertices;the remainingedgeshavebothendsamongthedi+1vertices:ofthese,letyihaveoneendinthe Multiply(7)by 1(reversingtheinequality)andadditto(6)toobtain: independentsetandoneoutside,andzihavebothendsoutside.thenwehave: xi+yi+2zidi(di+1) ki(di+1 ki) xi+2(yi+zi)di(di+1); yiki(di+1 ki); 2! ki(di+1 ki) (7) (6) Sincethenumberofedgesdeletedinthei-thstepispreciselyxi+yi+zi,wehavethefollowing extensionof(5): jejtxi=1 = di+1 2!+ di+1 2!+ ki 2!+ 2!+ di+1 ki 2! zi 2!: 6 (8)

7 independentsettwice. entailscountingedgesincidentontheindependentsetverticesoncebutthosefullyoutsidethe Wealsocountthetotaldegreeofverticesoutsidethemaximumindependentset,which Tosimplifytherighthandside,weaddPti=1(di+1)+ki+(di+1 ki)andcompensatefor Nowaddtwice(8)andtwice(9)toobtain: 2(n )txi=1di(di+1)+ki(ki 1)+(di+1 ki)(di ki): (n )txi=1zi+jej: (9) thisbyadding2ntothelefthandside(invoking(2)).then: Theclaimfollowsfromthis. Using(2),(4)andtheCauchy-Schwarzinequality,weobtain: 2(n )+2ntXi=1(di+1)2+k2i+(di+1 ki)2: 2((1 )+1)n[1+2+(1 )2]n2=t: (10) boundoftheorem2inthedenominatoroftheperformanceratiofunction,usetheidentity ThefollowingboundontheperformanceofGreedyonsparsegraphsfollowseasily.Weusethe =ninthenumerator,andobservethattheratioismaximizedwhen=1. Corollary4Grd+2 3.2Performanceguarantees chosenandexactlykverticesoftheindependentsetwereremoved.moreprecisely,ford= 0;1;:::;andk=0;1;:::;max(d;1),wedenetd;k=jfi:di=dandki=kgj:Withthis quiteyieldsourmainclaimabouttheperformanceratioofgreedy.wenowproceedtoanalyze theperformanceofgreedyusinganerscalpel. Lettd;kbethenumberofreductionsperformedbyGreedywhereavertexofdegreedwas Forbounded-degreegraphs,thegeneralexpressionobtainedinTheorem3almostbutnot 2 notation,wemayrewritetheconstraints(2),(4)and(10)as: Xd;k(d+1)td;k=n; Xd;kktd;k=; (11) ForthisweusethemethodofmultipliersdescribedinChv tal[6,page54]. Wewishtoextractfromtheseconstraintsthebestpossiblelowerboundfort=Pd;ktd;k. Xd;k[(d+1)2+k2+(d+1 k)2]td;k2n(+1) 2: (13) (12) Proof.Weneedtoconsidertwocasesbasedonthevalueof. Theorem5Gr(+2)=3 7

8 theconstraints(11)and(13),withmultipliers2(+1)and 1respectively,andobtain: Case0;1(mod3)[i.e.+11(mod3)].Weconstructthelinearcombinationof Let Xd;k[2(d+1)(+1) ((d+1)2+k2+(d+1 k)2)]td;k2: (d)=min C(d)=2(d+1)(+1) (d): k[(d+1)2+k2+(d+1 k)2] (14) Settd=Pktd;kandconcludefrom(14)that: XdC(d)td2; asrequired.itremainsonlytoestablishthefollowingclaim. Weshowbelowthat,ford=0;1;:::;,C(d)2(+2)=3.Thatwith(15)thengives +2 3t; (15) Claim:Ford=0;1;:::;,C(d)23(+2): Proofofclaim:Itcaneasilybeveriedthat: Note: Letf0:<!<andf1:<!<bedenedby: f0(x)=2x(+1) 32x2;f1(x)=f0(x) 12: (d)=(32(d+1)2ifdisodd 32(d+1)2+12ifdiseven concavefunctionsthatachievetheiruniquemaximumat^x=2(+1)=3.sincef0andf1are establishtheclaim,itisenoughtoverifythatf0atthenearestevenintegerto^xandf1atthe nearestoddintegerto^xareatmost2(+2)=3. polynomialsofdegree2inx,f0(^x+)=f0(^x )andf1(^x+)=f1(^x ),forall.thus,to Now,f0(x);f01(x)=0ix=2(+1)=3,andf00 C(d)=(f0(d+1)ifdisodd f1(d+1)ifdiseven to^xis2mandthenearestoddintegerto^xis2m+r.pluggingin,wendthat Let+1=3m+r,wherer=1(since+1=1(mod3)).Thenearesteveninteger 0(x);f00 1(x)= 3.Thus,bothf0andf1are establishingtheclaim. f0(2m)=f1(2m+r)=6m2+4mr=23(+2); 8

9 Case 1(mod3).Thistime,weconstructalinearcombinationoftheconstraints(11), (12)and(13),withmultipliers2(+1),2and 1respectively,andobtain: where C(d;k)=2(d+1)(+1)+2k (d+1)2 k2 (d+1 k)2 =2(d+1)( d)+2k(d+2 k): Xd;kC(d;k)td;k2(+1); (16) Itcanbeveriedthat thefunctionsf0:<!<andf1:<!<denedasfollows. LetC(d)=maxkC(d;k).ToobtaintheanupperboundonC(d)intermsof,weconsider f1(x):=2(x+1)( x)+12(x+2)2=x(2 32x)+2(+1) f0(x):=2(x+1)( x)+12(x+2)2 12=(x+1)(2 32x+32) in,wendthat ix=2=3andf00 integerto^x(i.e.(2 1)=3),andf1atthenearestevenintegerto^x(i.e.2(+1)=3).Plugging Sincef0andf1arepolynomialsofdegree2inx,itisenoughtoboundf0atthenearestodd Wewishtoboundf0(d)foroddvaluesofd,andf1(d)forevenvaluesofd.Now,f0(x);f01(x)=0 0(x);f00 1(x)= 3;thusf0andf1attaintheiruniquemaximumat^x=2=3. C(d)=(f0(d)ifdisodd f0(2 1 3)=( )(2 2 1 f1(d)ifdiseven f1(2(+1) 3)=2(+1) =23(+1)(+2); =23(+1)( 1+3) 3(2 (+1))+2(+1) 2+32) Thus,C(d)23(+1)(+2),andusing(16)weobtain+2 Theperformanceratiosprovedabovecannotbeimproved. 3.3Limitations =23(+1)(+2): Theorem6Gr+2 3 O(2=n);forevery3: 3t. whiletheconnectionsbetweentheindependentsetandthecliqueofthefollowingpairmissonly Proof.Wegiveadetailedconstructionfor1(mod3).Considerthefollowingfamilyof followedbyanindependentseton`vertices.thetwosubgraphsarecompletelyconnected, graphsh`,`2.wehaveachainofrepetitionsofapairofsubgraphs:acliqueon`vertices 9

10 asingleperfectmatching(i.e.eachvertexintheindependentsetisadjacentto` 1verticesin thefollowingclique).thechainendswithoneadditionalclique. showninblackandthemaximumindependentsetverticesingrey. Aninstanceofthisgraphwith`=3isshowninFigure1,withtheverticespickedbyGreedy Theessentialpropertyofthegraphisthatthedegreeoftheindependentsetverticesequals Figure1:InitialportionofahardgraphforGreedy,=7 willpickoneoftheverticesfromtherstcliqueandremovetheremainingverticesfromthe pair,reducingthegraphtoanidenticalchainwithonefewerpairs. thedegreeoftheverticesoftherstcliqueofthechain.wecanthereforeassumethatgreedy Torelatethattothedegreemeasures,wehavethat=3` 2,and of(n `)=2`+1.Theoptimalsolutioncontainsalltheindependentsetverticesforatotalof (n `)=2.ThisyieldsaratioofGr(H`)` 2`2=n: Thus,Greedyselectsonevertexfromeachpair,plusonefromthenalclique,foratotal Thus, and d2 Gr+2 `2!+`2+`(` 1)!=2`=5` 3 3 O(2=n) 2: evenwhen1=2. Gr2d+3 5 O(d2=n) (17) For0(mod3),theelementsareoftheform: Forthecaseof0;2(mod3)weneedmorecomplicatedchainsofgroupsofsixsubgraphs. K` 1 K` K` 1 K` K` K` 1 (18) clique. ratherthantotheoneimmediatelyfollowing.thechainisnishedwithanadditional2` 1 whereks(ks)denotesaclique(independentset)onsvertices,respectively.inallcases,a cliqueiscompletelyconnectedtothefollowingindependentset,whileconnectionsfromthe therstindependentset(thesecondsubgraph)gotowardstherstcliqueinthenextgroup independentsettothenextcliquemissasingleperfectmatching.inaddition,` 1edgesfrom 10

11 setis3` 1pergroup.Hence,ignoringtheendofthechain,theapproximationratiois theleft-mostremainingclique.hence,wemayassumethatgreedywillpickexactlyonevertex perclique,or3nodespergroup,whilethenumberofverticesinthemaximumindependent Thegraphsaredesignedsothatminimumdegreewillstayas2` 2andequalthedegreeof Anexamplefor`=3isgiveninFigure2. Figure2:InitialportionofahardgraphforGreedy,=6 reader. andedgesgoingfromthesecondtothefthsubgraph.weleavethedetailstothecurious ` 1=3=(+2)=3,sincemaximumdegreeis3` 3. lary4)isoptimalinanasymptoticsense. Wenowshowthattheboundontheperformanceratiointermsofaveragedegree(Corol- Thecaseof2(mod3)issimilar,witheachelementoftheform: Theorem7Grd+2 K` 1 K`+1 K` K` K` K` independentseteachofwhosenodesareadjacenttothe` 1followingsinglevertexcliques. thecliquesreducedtosinglevertices.eachpairconsistsofavertexadjacenttoa`-element Thechainthenendswithasingle`-cliquethattheverticesofthelast` 1independentsets ratioholds.thegraphsconsistsofchainsofpairsofsubgraphsasinthepreviousexamplewith areadjacentto.anexampleisgiveninfigure3. Proof.Foreachvalueof,wedescribeaninnitefamilyofgraphsforwhichtheclaimed 2 O(1=d) thechain,theaveragedegreeis2`2=(`+1)=2` 2+2=(`+1)andtheratioobtainedis: Thereare`2edgesforeachpairconsistingof`+1vertices.Thus,ifweignoretheendof Figure3:HardgraphforGreedyintermsofaveragedegree TheendofthechainincreasestheaveragedegreebyroughlyO(`2=n),whichdisappearsinto thelowerorderterm. `d d+2:

12 theindependentsets,possiblybyconnectingeachindependentsettoseveralsubsequentcliques. Weomitthedetails. 4Greedywithpreprocessing forarangeofvaluesof.thisinvolvesvaryingthesizeofthecliquesrelativetothesizeof VariationsoftheaboveconstructionsshowthattheboundsofTheorems2and3aretight FindingamaximumindependentsetofagraphG=(V;E)canbeformulatedasaninteger Iftheintegralityconditiononthesolutionsisdropped,weobtainthefractionalindependent programmingproblemwhichmaximizespxioverthe0-1solutionsofthesystemoflinear setproblem,orthelinearprogrammingrelaxationoftheindependentsetproblem.asshown byedmondsandpulleyblank(see[24]),thiscanbesolvedecientlyviaabipartitegraph inequalities: constructedasfollows:formtwocopiesofthevertexsetv,andletverticesindierent xi+xj1foreach(vi;vj)2e: xi0foreachvi2v copiesbeadjacentitheycorrespondtoadjacentverticesing.givenacharacteristicvector (y1;y2;:::;yn;y01;y02;:::;y0n)ofamaximumindependentsetofthisbipartitegraph,anoptimal LPsolutionofGisgivenby: verticesinagiven(arbitrarybutxed)lpsolution.nemhauserandtrotter[24]showedthat BytheK nig-egervarytheorem[20,p.90],theindependencenumberofabipartitegraphequals hencethelprelaxationoftheindependentsetproblem. thenumberofverticeslessthenumberofedgesofamaximummatching.thebipartitematching problemcanbecomputedintimeo(pnjej)byanalgorithmofhopcroftandkarp[15],and WecanpartitionVintothesetsOne,Half,andZero,correspondingtothevaluesofthe xi=12(yi+y0i): performanceofapproximateindependentsetalgorithms.computeanlpsolution,andapply thesetoneiscontainedinsomeoptimalindependentsetoofg.furthermore,zerocannot isgreaterthanthat. becontainedino,asotherwisethevalueofthoseverticescouldbeunilaterallyincreased. solution,orn=2,sothisapproachisparticularlyvaluablewhentheindependencenumberofg theapproximationalgorithmonlyonthegraphhinducedbyhalf.tothesolutionfound bythealgorithm,wecannowaddtheverticesofone,possiblyresultinginaconsiderable improvement.theindependencenumberofhcanbeatmostthesumofthevaluesofthelp Basedontheseresults,Hochbaum[14]proposedapreprocessingmethodtoimprovethe andinh,theindependencefractionisatmost1=2. andtheoptimalsolutionconsistofonealongwiththerespectivesolutionsonthegraphh, obtaineddirectlyfromtheorem3byarguingthat1=2.thisisbecauseboththeapproximate Proof.Theupperboundof(+2)=3alreadyfollowsfromTheorem5,buthereitcanalsobe 1(mod3). Theorem8Greedywithpreprocessingattainsaratioof(+2)=3,whichistightwhen proofoftheorem6.theyhavethepropertythat,foranyindependentseti,thenumberof neighborsoftheverticesiniexceedsthesizeofi.itthenfollowsfromtheorem4of[24]that anylpsolutionhasallvaluesequaltohalf,i.e.preprocessingisofnohelp,soh=g. For1(mod3),thetightnessofthisratioisdemonstratedbythegraphsgiveninthe 12

13 Forinstance,wehaveatightratioof3=2when=3.Also,for=5wegetaratioof 16=72:286,downfromthe2:33promisedbyTheorem5,whilethereisagraphthatforcesa 4.1Averagedegree Agreatercareisneededforaresultintermsofaveragedegree.Oncepreprocessinghasbeen ratioof2:27.weomitthedetails. Wecanimproveonthisboundslightlywhen0;2(mod3)to(+2)=3 1=(3+2). Theorem9Gr+Pre2d+3 followherleadtoobtainasimilarresultasexpectedfromtheorem2. theboundsprovedongreedy.nevertheless,acloserlookshowsthattheboundswillcomplementeachotherashopedfor.hochbaum[14]showedthatthetur nboundongreedycanbe applied,theaveragedegreemayhavechangedfortheworseandwecannotimmediatelyapply complementedwiththe1=2promisetoyieldaperformanceratioboundof(d+1)=2.we solutionvalueissmallerthanifhalf=v.thesecondpropertyisthatthenumberofedgesin Proof.Letd0,0,n0denotetheaveragedegree,independencefraction,andnumberofvertices Gisatleastd0 ofhrespectively.recallthathisthesubgraphinducedbyhalf,andnotethat01=2. n0ispositive.thus,theaveragedegreeofgisboundedby: Following[14],weusetwoproperties.TherstisthatjOnejjZeroj,asotherwisetheLP 2n0+jOnej+jZeroj,sincewemayassumethatGisaconnectedgraphandthat 5 thesizeoftheoptimalsolution0n0+jonej.ourgoalisnowtoarguethat: Thesizeofthegreedysolutionwillbeatleast1+02 d0+1+0n0+jonej(25d0+35)n0+2jonej dd0n0+2jonej n0+2jonej: d0+1+0n0+jonej,invokingtheorem2,and and,ifjonej>0, fromwhichthetheoremfollows.cross-multiplying,wendthatthisentailsestablishing: (d0+1+0)(2d0+3)=5 n0+2jonej d d0+3 5: for0x1=2.thusitsucestoreplace0with1=2in(20)toobtain: (20),wemaynotethatthefunctionf(x)=2x+1 2(1+x2)=(d0+1+x)ismonotoneincreasing Ineq.(19)holdsbecausethelefthandsideismonotoneincreasingwith0for01.Toestablish 25 2d0+3+2d0+3 5 whichistrue,sincea=b+b=aisalwaysatleast2whena;barepositive. Thisagainistightfor1(mod3)byTheorem6,Ineq.(18). 13

14 degreebeselected,addedtothesolution,andremovedfromthegraphalongwithitsneighbors. 5Parallelalgorithm Theminimum-degreegreedyalgorithmstipulatesthatineachstepavertexofgloballyminimum hassomeinterestingimplications.forone,itopensupthepossibilityofthedesignofheuristics Assuch,itlooksimpossibletoparallelize,andoerslittlefreedomforheuristicimprovements. Fortunately,thisisacasewheretheanalysisguidesustowardsthedesignofbetterand/ormore generalalgorithms. Greedy. usingsecondaryselectionrulesororderingheuristicswhileretainingtheperformanceratiosof AsobservedinSection2,itsucestoselectacriticalvertex,i.e.onesatisfying(1).This bytakingtheadjacencymatrixofgtothethirdpowertwoverticesareadjacenting3ifthey arewithindistancethreeing.theadjacencymatricesusedhereareassumedtocontain1'son approximationalgorithmattainingthesameperformanceratios.verticescanbeselectedin selectedandprocessedconcurrently. verticeswithdisjointandnon-adjacentneighborhoods(i.e.ofdistancefourorgreater)canbe parallelaslongastheselectionofonedoesn'taecttheabovecriteriafortheother.inparticular, Thissuggestsanaturalapproachtoaparallelalgorithm.LetG3denotethegraphobtained Anotherapplicationisastraightforwardderivationofaparallelaswellasadistributed ins.parallelgreedy(g) thediagonal.foravertexsubsets,letn(s)denotethesetofverticesadjacenttosomenode while(v(g)6=;) I MIS H W ; subgraphofg3inducedbyw fv2v(g):viscriticalg WeassumethePRAMmodel,withanerdistinctiondependingontheMISalgorithmused. end returni G I I[MIS G (MIS[N(MIS)) maximalindependentsetofh icantfractionoftheverticesmustsimultaneouslysatisfyproperty(1). [21]ṪhefollowinglemmaduetoAlonandSzegedy(privatecommunication)showsthatasignif- Weremarkthatthisalgorithmcanalsobeimplementedinthedistributedmodelofcomputation Lemma10Atleast4 abovebyd(v)( d(v))2=4,andsinceitisintegral,itmustdierfromzerobyatleastone Proof.LetDvdenotePw2N(v)d(w) d(v)2.weshallshowthat whichimpliesthelemma.asobservedbyshearer[26],ev[dv]=0.thevalueofdvisbounded 2+4nverticesarecritical. whennegative.thus, 1(1 Pr[Dv0])+2=4Pr[Dv0]0: v[dv0]4 2+4; 14

15 Theclaimnowfollows. Theorem11ParallelGreedyndsanindependentsetofsizeandperformancesatisfyingTheorems3and5intimeO(lognmin(poly()logn;(log)!))usingnprocessorsintheEREW model. Proof.Eachvertexaddedtothesolutionwillsatisfyproperty(1)regardlessoftheorderof algorithm. fromaxedvertexintwostepsisatmost2,wehave: removalofthesimultaneouslychosenvertices.hence,theresultsofthetheoremsapplytothis Thus,thenumberofdeletedvertices,thatisjMIS[N(MIS)j,isatleastn=Delta4.Hence, Lemma10,thesizeofWisatleastn=2.EveryvertexinWisreachable(inG)fromMIS[ N(MIS)bya(notnecessarilysimple)pathoflength2.Sincethenumberofverticesreachable Letusnowestimatethetimecomplexity,startingwiththenumberofiterations.From thenumberofiterationsisatmost4logn. carefulcountingshowsthatthenumberofroundsisatmost!. critical.eitheruisselectedorsomevertexwithindistance2ofu,andthuswithindistanceat most+1ofv.thereareatmost+1suchnodes,andhence,numberofrounds.amore Alsonoticethatforanyvertexvinthegraph,somevertexuofdistanceatmost 1is jmis[n(mis)j2jwjn=2: ofthegraphh.analgorithmofgoldbergetal.[9]ndsamaximalindependentsetintime O(log(H)(2(H)+logn))usingalinearnumberofprocessors. Theonlynon-trivialstepineachroundisthecomputationofamaximalindependentset allverticesofhighdegree. Lemma12Theorem2andCorollary4holdevenifwersteliminate(simultaneously)all wellboundedintermsoftheaveragedegreed.fortunately,thiscanbeattainedbyrstdeleting NotethatthisisO(logn)onconstantdegreegraphs. WhiletheabovealgorithmyieldsasolutionsatisfyingTheorem2,itstimecomplexityisnot ThecombinedtimecomplexityisthereforeboundedbyO((6+logn)(log)min(4logn;!)). Proof.Considerthesubgraphinducedbyverticesofdegreelessthan3d+4.Let(1 )n,d0,0 denotethenumberofvertices,averagedegree,andindependencefraction,respectively.thus, representsthefractionoftheverticesthatwereofhighdegree.thepropositionisthat verticesofdegreeatleast3d+4. LHS=1+02 d0+1+0(1 )n1+2 Theindependencenumberoftheremaininggraphisatleasttheoriginalvaluelessthenumber Atleast((3d+3)=2)nedgesaredeleted,so d02jej (6jEj=n+4)n n n d(1 3) 4 d+1+n=rhs: 1 : (21). ofdeletedvertices,andthus0.weshallconsidertwocases,dependingonthevalueof 15

16 theoremis1=(d+1).thus,weplugin for0,for: Case 1=(d+1).LHSisminimizedwhen0isatitsminimum,whichbyTur n's Case 1=(d+1).LHSisatleast (1+)(1 ) 4isatmost(1+)(1 3),since1.Hence,(22)isatleastRHS. Thenumeratorisatleast(1+2)(1 2)(1 ),whichisatleast(1+2)(1 3).Also, d(1 3) 4+(1+)(1 )n: (1+( )2)(1 ) d0+1(1 )n(1 )2 1 (1 3)(d+1)n=1++(42)=(1 3) (22) Since(23)isatleast(24),(21)holds. atmost Ontheotherhand,RHSismonotoneincreasingwith,sowhen1=(d+1)+,itsvalueis d+1+1=(d+1)+n (1=(d+1)+)2 d+1: n: (24) (23) anecientapproximationintermsofd. Corollary13ThereisaEREWparallelalgorithmthatndsanindependentsetsatisfying Theorem2andCorollary4intimeO(lognmin(poly(d)logn;d!))usingnprocessors. Bydeletingrstallhigh-degreevertices(inparallel)beforeapplyingParallelGreedy,weobtain arguein[12],thisapproachyieldsaperformanceratioof(2d+4:5)=5. thatnearlymatchestheboundoftheorem9istouseeitherthecomplementofamaximal matchingortheoutputoftheabovealgorithm,whicheverindependentsetislarger.aswe areknown[17],andthusthesameappliestotheorem9.anecientdeterministicapproach whichnodeterministicparallelalgorithmsareknown.however,randomizedparallelalgorithms Thepreprocessingmethodoflastsectionrequiresthesolutionofbipartitematching,for andtheanonymousrefereesforthoroughimportantcorrections. Vas kchv talforcommentsandhelpfuladvice,dorithochbaumforcommentsandcorrections, References AcknowledgmentsWethankNogaAlonandMarioSzegedyforkindlyprovingLemma10, [1]N.Alon,U.Feige,A.Wigderson,andD.Zuckerman.Derandomizedgraphproducts. [2]S.Arora,C.Lund,R.Motwani,M.Sudan,andM.Szegedy.Proofvericationandhardness ofapproximationproblems.inproc.33rdann.ieeesymp.onfound.ofcomp.sci.,pages ComputationalComplexity,1995. [4]P.BermanandM.F rer.approximatingmaximumindependentsetinboundeddegree [3]P.BermanandT.Fujito.Ontheapproximationpropertiesofindependentsetproblemin degree3graphs.infourthworkshoponalgorithmsanddatastructures,aug ,Oct graphs.inproc.fifthann.acm-siamsymp.ondiscretealgorithms,pages365371, Jan

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