The best known result for the problem is an SDP based log log d log approximation due to Halperin It is also known that no d log approximation exists assuming the Unique Games Conjecture We show the following two results i The natural LP formulatio ID: 57316
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O(d(loglogd)2=log2d).Ourproofoftheorem1.1isnon-algorithmicanddoesnotgivean~O(d=log2d)approximationalgorithm,evenifweallowsub-exponentialinnrunningtime.Inparticular,itisbasedonaresultofAlon[3]ontheexistenceoflargeindependentsetsinlocallysparsegraphs,whichinturnisbasedonanentropy-basedapproachofShearer[22].Theorem1.1suggeststhat~ (d=log2d)mightbetherightapproximationthresholdfortheproblem,atleastwhendisconstant.Oursecondresultgivesanalgorithmthatim-provesthepreviouslyknownapproximationbyamodest (loglogd)factor,althoughattheexpenseofsomewhathigherrunningtime.Theorem1.2.ThereisanalgorithmbasedonlogO(1)dlevelsofthemixed-hierarchy,thathasapproximationratioO(d=logd)andrunsintimenO(1)exp(logO(1)d).Whiletheloglogdimprovementisperhapsnotsointerestingbyitself,ourtechniquesmaybemoreinter-esting.TheimprovementinTheorem1.2isbasedoncombiningHalperin'sapproachtogetherwithanideausedbyAjtai,Erd}os,KomlosandSzemeredi[1]toshowthatKr-free,degree-dgraphshaveindependencenum-ber r(nloglogd=d)(i.e.an (loglogd)factormorethanthenaivebound).Specically,weusetheprop-ertiesofhierarchiestosimulatetheapproachofAjtaietal.[1]ontopofHalperin'salgorithmandcombineboththeirimprovements.BothTheorems1.1and1.2extendtothecasewhendistheaveragedegree(insteadofmaximum)usingstandardtechniques.However,forsimplicityweonlyfocusonmaximumdegreesettinghere.1.2OurTechniques.Werstgiveabriefoverviewofprevioustechniques,andthendescribeourmainideas.Let (G)denotethesizeofamaximumindependentsetinagraphG.Letdand ddenotethemaximumandaveragedegreeofG.Thenaivegreedyalgorithmimplies (G)n=(d+1)foreveryG.Infact,itimpliesthat (G)= (n= d),sincewecandeletetheverticeswithdegreemorethan2 d(thereareatmostn=2ofthem)andthenapplythegreedyalgorithm.Asthegreedyguaranteeistightingeneral(e.g.ifthegraphisadisjointunionofn=(d+1)copiesofthecliqueKd+1),thetrivialupperboundof (G)ncannotgiveanapproximationbetterthand+1andhencestrongerupperboundsareneeded.Anaturalboundisthecliquecovernumber (G),denedastheminimumnumberofvertexdisjointcliquesneededtocoverV.Asanyindependentsetcancontainatmostonevertexfromanyclique, (G) (G).Ramsey-theoreticapproaches.Lookingat (G)naturallyleadstoRamseytheoreticconsiderations.Onemusteithershowthatthegraphcanbecoveredbylargecliques,inwhichcase (G)issmallandthetrivialn=(d+1)solutiongivesagoodapproximation.Otherwise,if (G)islarge,thenthisessentiallymeansthattherearenotmanylargecliquesandonemustarguethatalargeindependentsetexists(andcanbefoundeciently).Forboundeddegreegraphs,awell-knownresultofthistypeisthat (G)= (nlogd=d)fortriangle-freegraphs[2,21](i.e.iftherearenocliquesofsize3).Aparticularlyelegantproof(basedonanideaduetoShearer[22]isin[5].Moreoverthisboundistight,andsimpleprobabilisticconstructionsshowthatthisboundcannotbeimprovedevenforgraphswithlargegirth.ForthecaseofKr-freegraphswithr4,thesit-uationislessclear.Ajtaietal.[1]showedthatKr-freegraphshave (G)= (n(log(logd=r))=d),whichim-pliesthat (G)= (nloglogd=d)forr logd.ThisresultwasthebasisoftheO(d=loglogd)approximationdueto[14].Shearer[22]improvedthisresultsubstan-tiallyandshowedthat (G)cr(logd=loglogd)(n=d)forKr-freegraphs.Howeverhisargumentisexistential(andusesanelegantentropybasedapproach)andwearenotawareofanyalgorithmicvariantsorapplicationsofthisresult.Removingtheloglogdfactoraboveisamajoropenquestion,evenforr=4.SemideniteProgramming.SDPsprovideatleastasstrongupperboundson as (G),asthewell-known#-functionofLovaszsatises (G)#(G) (G).TheO(dloglogd=logd)approximationsdueto[15,13]arebothbasedonSDPs.FordetailsonSDPs,andtheLovasz#-function,wereferthereaderto[11]Boundingtheintegralitygap.WedescribethemainideasbehindTheorem1.1.Forsimplicity,supposethatwehavead-levelmixedhierarchyformulationM(d)(insteadofanO(log4d)-levels).TherstobservationisthatifM(d)hasasolutionofvaluesay4nloglogd=logd),thenHalperin'salgo-rithmalreadyreturnsasucientlylargeindependentset(detailsinTheorem3.1below).Sowecanassumethattheobjectivevalueiso(nloglogd=logd).Forcon-creteness,letussupposethateachvertexcontributesexactly1=logdtotheobjective.Observethatad-levelSherali-Adamssolutionspec-iesavalidlocaldistributionoverindependentsetsforeverysubsetofverticesofsized.Fixavertexvandconsiderthelocal-distributionvoverindependentsetsinthesubgraphGjN(v)ofGinducedbytheneighbor-hoodN(v)ofv.Asxu=1=logdforeveryvertexu,eachsuchu2N(v)mustliein1=logdfraction(ac- 2.ForallS0S[n]withjSjt+1,thedistributionD(S0);D(S)agreeonS0.ThenYS=PD(S)[^i2S(yi=1)]isafeasiblesolutionforthelevel-tSheraliAdamsrelaxation.Conversely,foranyfeasiblesolutionfY0Sgforthelevel-(t+1)Sherali-Adamsrelaxation,thereexistsafamilyofdistributionssatisfyingtheaboveproperties,aswellasYS=PD(S)[^i2S(yi=1)]=Y0SforallS0S[n]suchthatjSjt+1.Here,Condition1impliesthatforasubsetofverticesSwithjSjt+1,thelocal-distributionD(S)hassupportonthevalidindependentsetsinthegraphinducedonS,andCondition2guaranteesthatdi erentlocaldistributionsinduceaconsistentdistributiononthecommonelements.Forourpurposes,wewillalsoimposethePSDconstraintonthevariablesyijattherstlevel(i.e.weaddtheconstraintsin(2.3)onyijvariables).WewillcallthistheleveltmixedhierarchyformulationanddenoteitbyM(t).Asolutiontothe`-levelmixedrelaxationabovespeciesvaluesySformulti-setsSwithjSj`+1.HowevertokeepthenotationconsistentwiththeLP2.1,wewillusexitodenotethemarginalsonthesetscorrespondingtosinglevertices.Ramsey-theoreticlowerboundsontheindepen-dencenumber.Wewillusethefollowingresultonindependencenumberofgraphswithfewtriangles.Theorem2.2.([2,21])LetGbeagraphwithaveragedegree d,andatmost d2ntriangles.Then, (G)= (log(1=)(n= d)).Remark:Thisresultisalgorithmic.Infactitfollowsdirectlyfromtheclassicfactthat (G)= (nlog d= d)fortrianglefreegraphs[2,21].Indeed,sampleeachvertexwithprobabilityp=1=(2p d).Thisgivesagraphwithn0=pnverticesinexpectation(andtightlyconcentratedaroundthisvalue).Moreovertheaveragedegreeisd0=p d=1=2p andthenumberoftrianglesisp3 d2n=pn=4inexpectation.RemovingeveryvertexinvolvedinatrianglegivesatrianglefreegraphG0with (n0)vertices,and (G0)= (n0log(d0)=d0)= (nlog(1=)= d).Thefollowingresultwillbecrucialinourargu-ments.Theorem2.3.(Alon[3],Theorem1.1)LetG=(V;E)beagraphonnverticeswithmaximumde-greed1inwhichforeveryvertexv2Vthein-ducedsubgraphonthesetofallneighborsofvisk-colorable.Then,theindependencenumberofGisatleastc log(k+1)n dlogd,forsomeabsolutepositiveconstantc.Theresultabovealsoholdsunderthefollowingweakercondition.Theorem2.4.(Alon[3])LetGbeagraphwithmax-imumdegreed,andletk1beaninteger.IfforeveryvertexvandeverysubsetSN(v)withjSjklog2d,itholdsthatthesubgraphinducedonShasanindependentsetofsizeatleastjSj=k,then (G)= (nlogd=(dlogk)).Theproofoftheorem2.4canbefoundintheAppendix.3IntegralityGapInthissectionweshowthattheintegralitygapoftherelaxationM(log4d)isO(d(loglogd)2=log2d)=~O(d=log2d).GivenagraphGonnvertices,letYdenotesomeoptimumsolutiontotherelaxationM(log4d).WewillrstpreprocessthesolutionYslightlytohavesomedesirableproperties.3.1PreprocessingLetsdenotetheobjectivevalueofthesolutionY.Wecanassumethats4n=log2d,otherwisethenaivegreedyalgorithmtriviallygivesaO(d=log2d)approximation.Moreimportantly,wecanalsoassumethatsisnottoolarge,otherwiseHalperin'salgorithmalreadygivesagoodapproximation.Inparticular,weneedthefollowingresultabouttheSDPformulation(2.2).Theorem3.1.(Halperin[15],Lemma5.2)Let2[0;1 2]beaparameterandletZbethecollectionofvectorsvisatisfyingkvik2intheSDPsolution.Thenthereisanalgorithmthatreturnsanindependentsetofsize d2 dp lndjZj.AsHalperinusesanSDPwhichlooksdi erentfromours(2.2)(thoughtheyareequivalent),wesketchtheproofofTheorem3.1intheAppendixforcompleteness.NotethatintheformulationM(log4d),wehavekvik2=yii=y0i=xi.Let:=3loglogd=logdandletZdenotethesetofverticeswithxi.Then,theorem3.1returnsanindependentsetofsize (jZjlog5d=d).IfjZjn=log3dthenthisgivesanindependentofsize (nlog2d=d)andhenceanO(d=log2d)approximatesolution.SowecanassumethatjZjn=log3d.AsthecontributionofverticesinZtotheobjectivescanbeatmostjZj,thiscontributioniso(s)ass4n=log2d.Thisallowsustoupperboundsasfollows.(3.4)sn+jZj1=(3+o(1))nloglogd logd: withthegraphG0=GandapplythealgorithminLemma4.1repeatedlyuntilthecase(1)holdsorun-til`=(4loglogd)=iterationsarecompleted.LetG1;G2;:::;Gkbethesequenceofthesubgraphspro-ducedforsomek`.Letniand didenotethenumberofverticesandaveragedegreeofGi.Notethatby(4.7),ni= diformsanincreasingsequence.Iftheprocessterminatesafter`steps,thenn` d`1+1 `n d=n dlog (1)dandhencethegreedyalgorithmappliedtoG`givesthedesiredindependentset.Ontheotherhand,iftheprocessterminatesatstepkfork`,thenwegetanindependentset(incase1)ofsize ((nk= dk)loglogd).Thisisatleast ((n0= d0)loglogd),asni= diisincreasing.Asn0=nand d0= d,theresultfollows.ItremainstoproveLemma4.1.Proof.(Lemma4.1).StartwiththegraphGandletV=V(G).IfthereisavertexvsuchthatthegraphinducedonitsneighborhoodN(v)hasatleast djN(v)jedges,removeS=fvg[N(v)fromG.ContinuetheprocessontheremaininggraphwithvertexsetVnSuntilnosuchvertexexistsorwhenthenumberofremainingverticesrstfallsbelown=2.Weconsidertwocasesdependingonwhentheprocessterminates.1.Ifmorethann=2verticesareleft,thenintheremaininggraphG,theneighborhoodN(v)ofeachvertexvcontainsatmost djN(v)jedges,andthusthetotalnumberoftrianglesisatmost(1=3) dPvN(v)=n d2=3.MoreoverasGhasatleastn=2vertices,itsaveragedegreeisatmost2 d.Thusbytheorem2.2,Gcontainsanindependentsetofsize (log(1=)(n= d))= (nloglogd= d).Thisgivesthedesiredset.2.Ifn=2orfewerverticeswereleft(theremustbeatleastn=2�d),considerthesetsS1;:::;STthatwereremovedfromGduringtheprocess,whereSi=vi[N(vi).AsthenumberofedgesineachsubgraphSiisatleast djN(vi)j= djSi�1j djSij=2:ThenumberofedgesinGthathavebothendpointsinthesetSiforsomei,isatleastTXi=1 djSij=2 d=2(n=2)jEj=2;wherejEjdenotesthenumberofedgesinG.NowconsiderthedistributionovertheindependentsetsinSideterminedbythelocal-distributionD(Si)givenbytheSherali-Adamssolution.Foreachi=1;:::;Twedothefollowing:SampleanindependentsetTiSiaccordingtotheproba-bilitiesdeterminedbyD(Si).Letp=1=logd.Foreachvertexv2Ti,sampleitindependentlywithprobabilityp=xv,andrejectitotherwise.Thisstepiswell-denedasxvpforeachvertexv2G.LetUiTidenotethesetofverticesthataresampled.Clearly,eachUiisanindependentsets,andeachvertexv2SiliesinUiwithprobabilityexactlyp.Forverticesv2(Vn(S1[:::[ST)),wesampleeachofthemindependentlywithprobabilityp.LetG0bethegraphinducedonthesampledvertices.TheexpectednumberofverticesinG0isnp.Moreover,asjSijd+1foreachi,thenumberofverticesistightlyconcentratedaroundnp(withstandarddeviationO(p npd)).Thecrucialpointisthatanedgee2Gwithbothitsend-pointsinsomeSihas0probabilityoflyinginG0andexactlyp2probabilityotherwise.ThustheexpectednumberofedgesinG0isjEj(1�=2)p2=n d(1�=2)p2=2(andtightlyconcentrated).Thisgivesusthatd0=dp(1�=2)pluslowerordertermswithhighprobability.Thusn0= d0isatleast(1+=4)n=dwithhighprobability,asdesired.WenotethatweonlyrequiretheShearli-AdamslocaldistributiononsubsetsofverticesthatlieinsomeneighborhoodN(v).Thus,therunningtimetocomputethesolutiontotherelaxationisnO(1)exp(d).Wenowshowhowthenumberoflevelsrequiredcanbereduced.4.3Reducingthenumberoflevels.Wesketchtheideatoreducethenumberoflevelstopolylog(d),anddefertherelativelystandarddetailstothefullversionofthepaper.Observethatintheproofoflemma4.1,wedonotnecessarilyrequireadistributionovertheindependentsetsTiofSi.Infact,anydistributionoversubsetsTiwithedgedensityaconstantfactorlessthanthatofSisucestoarguethatthegraphG0hasn0=d0(1+c)n=dforsomeconstantc.NowthecrucialpointisthatthegraphinducedonSiisquitedense.Ithasatmostdverticesandatleast djSijedges.Moreover,wecanassumethat dd=loglogd,otherwisethegreedyalgorithm(ap-pliedafterremovingverticesofdegreemorethan2 d)al-readyreturnsanindependentsetofsize (nloglogd=d).ThustheedgedensityofSiisatleast djSij=jSij2 foreveryvertexvandeverysubsetSN(v)withjSjklog2d,itholdsthatthesubgraphinducedonShasanindependentsetofsizeatleastjSj=k,then (G)= (nlogd=(dlogk)).Werstneedthefollowingbasicfact(seeLemma2.2in[3]foraproof).Lemma5.1.LetFbeafamilyof2xdistinctsubsetsofanx-elementsetX.ThentheaveragesizeofamemberofFisatleastx=(10log(1+1=)).Proof.LetWbearandomindependentsetofverticesinG,chosenuniformlyamongallindependentsetsinG.Foreachvertexv,letXvbearandomvariabledenedasXv=djv\Wj+jN(v)\Wj.SincejWjcanbewrittenasPvjv\WjandasitsatisesjWj(1=d)PvjN(v)\Wj(avertexinWcanbeinatmostdsetsN(v)),wehavethatjWj1 2dXvXv:Thustoshowthat (G)islarge,itsucestoshowthat(5.8)E[Xv]logd=(80logk+1):Toshow(5.8),weprovethatinfactitholdsforeveryconditioningofthechoiceoftheindependentsetinV�(N(v)[fvg).Inparticular,letHdenotethesubgraphofGinducedonV�(N(v)[fvg).ForeachpossibleindependentsetSinH,wewillshowthatE[XvjW\V(H)=S]logd=(80logk+1):FixachoiceofS.LetXdenotethenon-neighborsofSinN(v),andletx=jXj.Letbesuchthat2xdenotesthenumberofindependentsetsintheinducedsubgraphGjXofGonX.Now,conditioningontheintersectionW\V(H)=S,thereareprecisely2x+1possibilitiesforW:oneinwhichW=S[fvgand2xinwhichv=2WandWistheunionofSwithanindependentsetinGjX.Bylemma5.1,theaveragesizeofanindependentsetinXisatleastx 10log1=+1andthuswehavethatE[XvjW\V(H)=S](5.9)d1 2x+1+x 10log(1=+1)2x 2x+1:Now,if2x+1p d,thenthersttermisatleastp dandwearedone.Otherwise,itmustbethatx(1=2)logdandhencetherighthandsidein(5.9)isatleast(5.10)logd 40log(1=+1):Wenowconsidertwocasesdependingonthevalueofx.Ifxklog2d,thenbyourassumption,XcontainscanindependentsetofsizeatleastjXj=kandhence2x2x=kandhence1=k.Thisgivesthat(5.10)isatleastlogd 40log(k+1).Ontheotherhandifxklog2d,thenbyourassumption,Xcontainsatleast2log2dindependentsets,andhencexlog2d.Asxd,itfollowsthatlog2d=d1=dandhencelog(1=+1)2logd.Thustherighthandsideof(5.9)isatleastx 20log(1=+1)log2d 40logd:5.2ProofofTheorem3.1.Weprovethefollowingresult.Theorem3.1[Halperin[15]]Let2[0;1 2]beaparameterandletZbethecollectionofvectorsvisatisfyingkvik2intheSDPsolution.Thenthereisanalgorithmthatreturnsanindependentsetofsize d2 dp lndjZj.Proof.Letai=viv0=kvik2,andletwi=vi�hvi;v0iv0denotetheprojectionofvitov?0,thehyper-planeorthogonaltov0.Askwik2+hvi;v0i2=kvik2,weobtainkwik2=ai�a2i.Letui=wi=jwij.Nowforanypairofverticesi;j,wehavethatwiwj=vivj�hvi;v0ihvj;v0i:Asvivj=0if(i;j)2E,wehavethatwiwj=vivj�hvi;v0ihvj;v0i=�aiajandhenceuiuj=�p aiaj p (1�ai)(1�aj)� (1�):Thelaststepfollowsasai;ajandasx=1�xisincreasingforx2[0;1].Thustheunitvectorsuicanbeviewedasafeasiblesolutiontoavectork-coloring(inthesenseof[17])wherekissuchthat1=(k�1)==(1�).Thisgivesk=1=,andnowwecanusetheresultof[17](Lemma7.1)thatsuchgraphshaveindependentsetsofsize (jZj=d1�2=kp lnd)= (jZjd2=(dp lnd)).