log3n UFEIGEMLANGBERGANDGSCHECHTMANsubjectto ijCOLMinimizesubjectto ij ijThefunctionCOListhevectorchromaticnumberofasde ID: 828489
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1 log3n U.FEIGE,M.LANGBERG,ANDG.SCHECHTMAN
log3n U.FEIGE,M.LANGBERG,ANDG.SCHECHTMANmaximumdegree,thenforeverypositiveintegeritalsoholdsforsomegraphverticesandthesamemaximumdegree.(Simplymakedisjointcopiesoftheoriginalgraph.)Hencethetheorembecomesstrongerasbecomeslargerincomparisonto.Weguaranteethatcanbetakenatleastaslargeasforsomethatdependson.Foreveryxed,thevalueofcanbetakenasaxedconstantboundedfrom0andindependentof.WehavenotmadeanattempttondthetightestpossiblerelationbetweenProoftechniques.Thegraphsthatweuseareessentiallythesamegraphsthatwereusedin[FS02]toshowintegralitygapsforsemideniteprogramsforMax-Cut.Namely,theyareobtainedbyplacingpointsatrandomona-dimensionalunitsphereandconnectingtwopointsbyanedgeiftheinnerproductoftheirrespectivevectorsisbelow1).Suchgraphsarenecessarilyvector-colorable,astheembeddingonthesphereisavector-coloring.Sothebulkoftheworkisinprovingthattheyhavenolargeindependentset.Forthisweuseatwo-phaseplan.Firstweconsideracontinuous(innite)graph,whereeverypointonthesphereisaverte
2 xandtwoverticesareconnectedbyanedgeifthe
xandtwoverticesareconnectedbyanedgeiftheinnerproductoftheirrespectivevectorsisbelow1).Onthiscontinuousgraphweusecertainsymmetrizationtechniquesinordertoanalyzeitsproperties.Specically,weprovecertaininequalitiesregardingitsexpansion.Inthesecondphase,weconsidera(nite)randomvertexinducedsubgraphofthecontinuousgraph.Basedontheexpansionpropertiesofthecontinuousgraph,weshowthattherandomsamplehasnolargeindependentset.Tworemarksareinorderhere.Oneisthatitisveryimportantforourboundsthatthenalrandomgraphdoesnotcontaintoomanyverticescomparedtothe.Asmallnumberofverticesimplieslowdegree,andallowsforamorefavorablerelationbetweenthemaximumdegreeandthechromaticnumber.Forthisreasonwecannotusethecontinuousgraph(oranitediscreteapproximationofthecontinuousgraph)asis.Itsdegreeisverylargecomparedtoitschromaticnumber.Wethusconsiderthegraphobtainedbyrandomlysamplingtheverticesofthecontinuousgraph.Insection7,wefollowasuggestionofLucaTrevisan(privatecommunication)andpresentanalternativepro
3 ofofTheorem1.2(1)byconsideringthegraphob
ofofTheorem1.2(1)byconsideringthegraphobtainedbyrandomlysamplingtheedgesofthecontinuousgraph.Itappearsthatparts(2)and(3)ofTheorem1.2cannotbeprovenusingedgesampling.Theotherremarkisthatwedonotgetanexplicitgraphasourexample,butratherarandomgraph(oradistributionongraphs).Thisistosomeextentunavoid-able,giventhattherearenoknownecientdeterministicconstructionsofRamseygraphs(graphsinwhichthesizeofthemaximumindependentsetandmaximumcliquearebothboundedbysomepolylogin).Thegraphsweconstruct(when=logloglog)areRamseygraphs,becauseitcanbeshownthatthemaximumcliquesizeisneverlargerthanthevectorcoloringnumber.Propertytesting.ThefollowingprobleminpropertytestingisaddressedbyGol-dreich,Goldwasser,andRon[GGR98].Forsomevalueof1,consideragraphwiththefollowingpromise:eitherithasanindependentsetofsize,oritisfarfromanysuchgraph,inthesensethatanyvertex-inducedsubgraphofverticesinducesatleastedges.Onewantsanalgorithmthatsamplesasfewverticesaspossible,looksatthesubgraphinducedonthem,and,base
4 donthesizeofthemaximuminde-pendentsetint
donthesizeofthemaximuminde-pendentsetinthatsubgraph,decidescorrectly(withhighprobability)whichofthetwocasesabovehold.In[GGR98]itisshownthatasampleofsizeproportionaltosuces.Weareinasomewhatsimilarsituationwhenwemovefromthecontinuousgraphtoourrandomsample.Thecontinuousgraphisfarfromhavinganindepen- U.FEIGE,M.LANGBERG,ANDG.SCHECHTMANsubjectto i,jCOL)Minimizesubjectto i,j i,jThefunctionCOL)isthevectorchromaticnumberofasdenedin[KMS98].ThefunctionCOL)isthestrictvectorchromaticnumberofandisequaltotheLov´functionon[Lov79,KMS98],whereisthecomplementgraphof.Finally,thefunctionCOL)willbereferredtoasthestrongvectorchromaticnumberasdenedin[Sze94,Cha02].Let)denotethesizeofthemaximumclique;inthefollowingweshowthatCOLCOLCOLItisnothardtoverifythatCOLCOLCOL).Toshowtheotherinequalitiesweneedthefollowingfact.Foreveryinteger,theunitvec-thatminimizethevalueofmaxmax,...,k]vi,vjaretheverticesofasimplexincenteredattheorigin.Foreachi,jj,...,k],thesevectorssatisfy NowtoprovetheinequalityC
5 OL),considera.Thepartitionsthevertexseti
OL),considera.Thepartitionsthevertexsetintocolorclasses.Assigningeachcolorclasswiththecorrespondingvectorabove,weobtainavalidassignmentCOL).ToshowthatCOL),consideragraphwithmaximumcliquesize).ToobtainavalidassignmentofvectorstoCOL)ofvaluerequirethatallpairsofvectorscorrespondingtotheverticesofthemaximumcliquewillhaveinnerproductofvalueatmost .Asmentionedabove,thiscanhappenonlyifRemarkTheresultsofourworkshowalargegapbetweenCOL)and)(Theorem1.2).Combiningtheseresultswithcertainprooftechniquesappearingin[Sze94],asimilargapbetween)andCOL)canalsobeobtained.Detailsareomitted.3.Thecontinuousgraph.bealargeconstant,andletbethe-dimensionalunitsphere.LetV,E)bethecontinuousgraphinwhich(a)thevertexsetconsistsofallthepointsontheunitsphereand(b)theedgesetconsistsofallpairsofverticeswhoserespectivevectors(fromtheorigin)formanangleofatleastarccos(1)).Asthesizeofisinnite(anduncountable),termssuchasthenumberofverticesinwillbereplacedbythecontinuousanaloguemeasureInthissectionweanalyzev
6 ariouspropertiesof.Specically,weshowtha
ariouspropertiesof.Specically,weshowthathascertainexpansionproperties.Wethenusethisfactinsection4toprovethemaintheoremofourwork.Inouranalysis,wewillassumethatthedimension(atleast)averylargeconstant(ourproofsrelyonsuch).AdditionalconstantsthatwillbepresentedintheremainderofthissectionaretobeviewedasindependentofDefinition3.1(spheremeasureLetbethenormalizednaturalmeasureon,andletbetheinducedmeasureon.Forany U.FEIGE,M.LANGBERG,ANDG.SCHECHTMAN vC(a,b) azN Fig.1Projectingontothetwo-dimensionalsubspace,weobtainthecircleabove.istheprojectionofan-capcenteredat(whereisthedistanceofthecapfromtheorigin).Thevertex isontheboundaryofistheprojectiononthesphereofthesetofpointsthatareadjacentto(i.e.,formanangleofatleast).Theshadedsectionisprojectionof).Thepointa,bistheclosestpointoftheprojectionoftotheorigin.Itisnothardtoverifythat=(11)+.Finally,wedenotethevalue .Claimaddressesthemeasureofandstatesthatitisessentiallythemeasureofa-cap.Thisisdonebystudyingthepointsinwhoseprojectionf
7 allsa,b.Roughlyspeaking,werstshowthatsu
allsa,b.Roughlyspeaking,werstshowthatsuchpointsarein;then,usingClaim,weshowthatthemeasureofthesepointsisessentiallythemeasureofa-cap.Proof.Let.Letbean-capcenteredat.W.l.o.g.wewillassume=(10).Consideravertexontheboundaryof.Letbethesetofverticesadjacentto.Westartbycomputingthemeasureofverticesthatareneighborsofandareinthecap,i.e.,themeasureofv,uClaim3.7.Leta,kbeasinTheorem.Let avertexontheboundaryof.Letbethesetofneighborsof.Let a2+(1/(k1)+a2)2 .Finallylet log( forasucientlylargeconstant.Themeasureofverticesin 21z2d1 2µ(N(v))1z2d1 Claim3.7addressesthemeasureof),andstatesthatitisessentiallythemeasureofa a2+(1/(k1)+a2)2 .ToproveClaim3.7westudythemeasureofcertainrestrictedsetsin.ThesesetsarestudiedinClaim9.7ofsection9.Claims3.7and9.7aredepictedinFigure1andproveninsection9.TocompletetheproofofTheorem3.6,leta,z,beasinClaim3.7.Foravertexbethesetofverticesadjacentto.Fortheupperbound,noticethatofallverticesin,theverticesinwhich)islargestaretheverticesontheboundar
8 yof.ByClaim3.7,forthesevertices)isbounde
yof.ByClaim3.7,forthesevertices)isboundedby(1 .Wethusconcludethat)isboundedbythemeasureof GRAPHSWITHTINYVECTORCHROMATICNUMBERSverticesintimes(1 .Thatis, 1)+ 1a2d1 Asforthelowerbound,let)beavertexinwithrstcoordinateofvalue.Consideranyvertexwithrstcoordinateofvaluelessthan.Itisnothardtoverifythatthemeasureofisgreaterthanthemeasureof.UsingananalysissimilartothatofClaim3.7,wehavethat)isgreaterthanorequalto 1)+ 1(a+)2d1 2(1c)d1 21z2d1 forasucientlylargeconstant(whichchangesvaluesbetweenbothsidesofthesecondinequality).Furthermore,forourchoiceof,themeasureofvertices)inisatleast2(Claim9.3).Hence,weconcludethat)isatleast 2(1c)d1 21z2d1 Simplifyingtheaboveexpression,weconcludeourassertion. Theorem3.6addressesthecaseinwhichisconstantandthecapsconsideredarebothofmeasure)foraconstantvalueof.FortheproofofTheorem1.2(2)wealsoneedtoaddressnonconstantvaluesofwhichdependonTheorem3.8.Let=(log( .Let1)=.Letandletbean-capcenteredat.Letbethevalueof.Thevalueofisint
9 herange poly k2a2d1 2,12(k1) k2a2
herange poly k2a2d1 2,12(k1) k2a2d1 TheoutlineoftheproofofTheorem3.8issimilartothatofTheorem3.6.Afullproofappearsinsection9.Definition3.9.Let.AgraphV,Eissaidtobepairwise,connectedieverytwo(notnecessarilydisjoint)subsetsofmeasureA,BCombiningTheorems3.5,3.6,and3.8weobtainthefollowingresult.Corollary3.10.Leta,kbedenedasinTheorem.Thegraphispairwise-connected.Roughlyspeaking,Corollary3.10addressestheexpansionpropertiesofthecon-tinuousgraph.Insection5,weshowthatthesepropertiesimplycertainupperboundsontheindependencenumberofasmallrandomsampleof.Namely,weprovethefollowingtheorem.Theorem3.11.Leta,kbedenedasinTheorem.Letbearandomsampleofverticesof(accordingtotheuniformdistributiononLetbeasucientlylargeconstant.If ,thentheprobabilityisatmostInthefollowingsection,Theorem3.11isusedtoprovethemainresultofthiswork,Theorem1.2.TheproofofTheorem3.11willbepresentedinsection5.2.4.ProofofTheorem1.2.Recallthatwearelookingforagraphforwhichboththevectorchromaticnumberandthesiz
10 eofthemaximumindependentsetare GRAPHSWIT
eofthemaximumindependentsetare GRAPHSWITHTINYVECTORCHROMATICNUMBERSClaim4.1, )withprobability4.ItislefttoanalyzethemaximumdegreeofClaim4.2.Withprobabilitygreaterthan,themaximumdegreeofthesubgraphisintherange Proof.Consideravertex.Letbethedegreeof.Aseveryvertexinisofdegree,theexpectedvalueofis(1).Thus(usingstandardbounds)theprobabilitythatdeviatesfromitsexpectationbymorethanaconstantfractionofitsexpectationisatmost2.Theprobabilitythatsomevertexinhasdegree ]isthusatmost2log(4forourchoiceof TherstassertionofTheorem1.2nowfollowsusingbasiccalculations. ProofofTheorem1.2(2).Letbealargeconstant.Let log().Let .LetV,E)bethecontinuousgraphfromsection3.Recall(Corollary3.10)thatispairwise-connected.Let).Thisimplies(Theorem3.11)thatwithprobability4arandomofsize )satisesisasucientlylargeconstant).AsbeforewestartbysimplifyingtheexpressionboundingClaim4.3.Thereexistsaconstants.t.withprobabilityatleastarandomofsize withprobabilityProof.Recallthat log( .Asbefore,itsucestobound).
11 ByTheorem3.8,wehave poly k2d1 Further
ByTheorem3.8,wehave poly k2d1 Furthermore,usingClaims9.1and9.2(ofsection9),weobtain k2d1 2=1a2 k2d1 21 poly polyWeconcludethatthereexistsaconstantsuchthat d. bearandomsubsetofverticesofofsize ).Noticethatlog( )and ).Bydenition,vectorcolorable.Thisimpliesthatanysubgraphof(includingthatinducedby)islog( log(log(vectorcolorable,whichcompletestheproofoftherstpartofourassertion.Forthesecondpartofourassertion,byClaim4.3thesubsetdoesnothaveanindependentsetofsize)withprobabilityatleast34(forsomedierentconstant ProofofTheorem1.2(3).Theprooffollowsthelineofproofappearingabove.Ingeneral,weusethegraph,butthistimethevalueofissettobe36.Again,ispairwise-connected,where)canbeboundedbyapproximately( 4(14a2))d1 .Let)and).ByTheorem3.11,arandomsubsetofsize )doesnothaveanindependentsetof(withprobability4).Computingthevalueoflog,weobtainourassertion.Wewouldliketonotethatresultsofasimilarnaturecanbeobtainedusingtheabovetechniquesforanyvalueof 5.Randomsampling.Wenowturnto
12 provingTheorem3.11statedinsec-tion3.This
provingTheorem3.11statedinsec-tion3.Thisisdoneintwosteps.Insection5.1weproveresultsanalogoustothosepresentedinTheorem3.11whenthegraphsconsideredarenite.Insection5.2we GRAPHSWITHTINYVECTORCHROMATICNUMBERSThen,usingthestandardunionboundonallsubsetsofsizegreaterthanweboundtheprobabilitythat�sThroughoutthissectionweanalyzethepropertiesofrandomsubsetswhichareassumedtobesmall.Namely,weassumethatthevalueofandtheparameters,andsatisfy(a) and(b)scnforasucientlysmallconstant.Inourapplications(andalsoinstandardones)theseassumptionshold.Insection5.1weanalyzetheaboveproofstrategyandshowthatitsucestoboundaconditionslightlyweakerthanthecondition&#x-277;&#x.800;s.Namely,usingthisscheme,weareabletoboundtheprobabilityforwhich&#x-277;&#x.800;sforsucientlylargeconstants.ThisresultisusedtoproveTheorem3.11ofsection3.Insection5.3wereneourschemeandobtainthemainresultofthissection.Theorem5.4.Letbea,-connectedgraph.Letbearandomsampleofofsize.Foranyconstantthereexistsaconstan
13 t(dependingonalone)s.t. 3log thenthep
t(dependingonalone)s.t. 3log thentheprobabilitythathasanindependentsetofsize�sisatmost ispairwise,-connected,and 3log log( thentheproba-bilitythathasanindependentsetofsize�sisatmostlog(1 5.1.Thenaivescheme.V,E)bea,-connectedgraph(pairwiseornot).Inthissectionwestudytheprobabilitythatarandomsubsetisanindependentset.Wethenusethisresulttoboundtheprobabilitythatarandomsubsetofsizehasalargeindependentset.Wewouldliketobound(fromabove)theprobabilitythatinducesanindepen-dentset.Letbetheverticesof.ConsiderchoosingtheverticesofonebyonesuchthatateachsteptherandomsubsetchosensofarisAssumethatatsomestageisanindependentset.Wewouldliketoshow(withhighprobability)thatafteraddingtheremainingverticesof,thenalsetwillnotbeanindependentset.)(forindependent)bethesetofverticesinwhicharenotadjacenttoanyverticesin,andlet)bethesetofverticesthatareadjacenttoavertex.Considerthenextrandomvertex.Ifischosenfromisnolongeranindependentset(implyingthatneitheris),andweviewthis
14 roundasasuccess.Otherwise,happenstobein)
roundasasuccess.Otherwise,happenstobein),andisstillanindependentset.Butifalsohappenstohavemanyneighborsin),thenaddingittowillsubstantiallyreducethesizeof),whichworksinourfavor.ThislatercaseisalsoviewedasasuccessfulroundregardingMotivatedbythediscussionabove,wecontinuewiththefollowingdenitions.Asbefore,letV,E)bea,-connectedgraph(pairwiseornot),letbeasetofverticesin,andlet.Eachsubsetdenesthefollowingpartition(,HI)ofbetheverticesthatarenotadjacenttoanyvertexin(noticethatitmaybethecasethatisnowpartitionedintotwoparts:verticesinwhichhavedegree,denotedastheset,andverticesofdegree,denotedas.Namely,isdenedtobetheverticesofwithminimaldegree(inthesubgraphinducedby),andisdenedtobetheremainingverticesof.Tiesarebrokenarbitrarilyorinverticesin(namely,verticesinareplacedinbeforeotherverticesofidenticaldegree).Ifitisthecasethat,thenisdenedtobe GRAPHSWITHTINYVECTORCHROMATICNUMBERSNowtoproveourlemma,considerthesubsetsandtheircor-respondingpartitions(,HI).Let.Let)betheverticesad
15 jacentto.Wewouldliketoboundthenumberofve
jacentto.Wewouldliketoboundthenumberofverticesthatarein.Westartwiththecaseinwhich,-connected.Consideravertex.ByClaim5.6,itsdegreein .Eachvertexincreasesthesizeofbyatleast.Initially,isempty,andafterchosen,.Weconcludethatthereareatmost verticeswhichare.Thisboundcanbefurtherimprovedbyafactorofapproximatelylog(1 usingtighteranalysiswhenisalsopairwise,-connected;detailsfollow.0beaninteger,andlet 2x+1,n Wewouldliketoboundthesizeofforallpossiblevaluesof.Westartbyconsideringvaluesofbetween0andlog(11.Consideravertexinwhich.Thatis, 2xi1|�n .ByClaim5.6,thedegreeof 22|Ii1 22n .Eachvertexinwhichincreasesthesizeofbyatleast.Forsuchvertices,isofsizeatleast atmost .Weconcludethatisofsizeatmost Forlog(1,thesetisasubsetofRecallthatwhenever.Thisimpliesthatinthesecases.Insum,weconcludethatlog(1 ,whichconcludesourproof. Theorem5.7.Letbea,-connectedgraph(pairwiseornot).LetbeasinLemma.Let.Theprobabilitythatrandomverticesofinduceanindependentsetisatmost Proof.Letbeasetofrand
16 omvertices.Asmentionedprevi-ously,thepro
omvertices.Asmentionedprevi-ously,theprobabilitythatisatmost.Thisfollowsfromthefactthat(1)thesizeofisatmost,(2)(byourdenitions),and(3)thevertexisrandominNowinorderfortobeanindependentset,everyvertexmustbeinthe.Furthermore,byLemma5.5allbutverticesmustsatisfyHence,theprobabilitythatisanindependentsetisatmost tkt=kek tt. bealargeconstant.WenowuseTheorem5.7toboundtheprobabilitythatarandomsubsetofsizehasanindependentsetofsize�s.TheresultisthefollowingCorollary5.8,whichwillbeusedinsection5.2toproveTheorem3.11.Insection5.3wereneourprooftechniquesandgetridoftheparameter.Thatis,weboundtheprobabilitythatarandomsubsetofsizehasanindependentsetofsize�sCorollary5.8.Letbea,-connectedgraph(pairwiseornot).LetasinLemma.Letbearandomsampleofofsize.Let�e,andletasucientlylargeconstant.Iflog(1 ,thentheprobabilitythat�satmost sProof.Lets.UsingTheorem5.7andthefactthatasubsetrandomin,theprobabilitythatthereisanindependentsetofsizeisat GRAPHSWI
17 THTINYVECTORCHROMATICNUMBERSbeanindepend
THTINYVECTORCHROMATICNUMBERSbeanindependentset,let,andlet,HI)bethepartition(asdenedinsection5.1)correspondingto.Roughlyspeaking,insection5.1,everytimeavertexwaschosen,thesubsetwasupdated.waschosenin,thengrewsubstantially,andifwaschoseninthesubsetwasonlyslightlychanged.Wewouldliketochangethedenitionofthepartition(,HI)correspondingtotoensurethatisalwayssub-stantiallysmallerthan.ThiscannotbedoneunlesswerelaxthedenitionofInournewdenition,willnolongerrepresenttheentiresetofverticesadja-centto;rather,willincludeonlyasubsetofverticesadjacentto(asubsetwhichissubstantiallysmallerthan).Namely,inournewdenitionofthepartition,HI)thesetischangedonlyifwaschosenin.Inthecaseinwhich,wedonotchangeatall.Aswewillsee,suchadenitionwillimplythat,whichwillnowsuceforourproof.Anewpartition.beasubsetof.Letbeasubsetofofsize.Eachsuchsubsetdenesapartition(,HI)of.Asbefore,let1.Initiallyistheverticesinofminimaldegree(in,and.Intheabove,tiesarebrokenbyanassumedorderingontheverticesin2.Let(,HI)
18 bethepartitioncorrespondingto,andletbean
bethepartitioncorrespondingto,andletbeanewrandomvertex.Let;thenwedenethepartition,HI).Let)betheneighborsof.Weconsiderthefollowingcases:,thenthepartitioncorrespondingtowillbeexactlythepartitioncorrespondingto,namely,.Noticethatthisimpliesthatnolongerrepresentsallneighborsof.Theremaybeverticesadjacenttowhichare,thenweconsidertwosubcases:,then,andaredenedasinsection5.1.Namely,isdenedtoisdenedtobetheverticesofminimaldegree(inthesubgraphinducedby),anddenedtobetheremainingverticesof.Tiesarebrokenbytheassumedorderingon,thenlet)betherst(accordingtotheassumedorderingon)(1verticesinandset).Furthermore,settobetheverticesof,andtobeempty.Noticethatinthiscase,isofsizeexactly(1then,onceagain,thepartitioncorrespondingtobeexactlythepartitioncorrespondingtoAfewremarksareinorder.First,itisnothardtoverifythatthedenitionaboveimpliesthefollowingclaim.Claim5.9.Let.Thepartitions,HIcorrespondingtoasdenedabovesatisfythatthesetistheverticesofminimaldegreeinSecond,duetotheiterativedeni
19 tionofournewpartition,thepartitions()cor
tionofournewpartition,thepartitions()correspondingtothesubsetsdependstronglyonthespecicorderingof GRAPHSWITHTINYVECTORCHROMATICNUMBERSWenowaddresstheprobabilitythatarandomsubsetisamaximumfreeset.Wewillthenusetheunionboundonallsubsetsofsize�stoobtainourresults.Lemma5.14.Letbea,-connectedgraph(pairwiseornot).LetbeasinLemma.Let.LetbeanorderedrandomsampleofofsizeTheprobabilitythatagivensubsetisamaximumfreesetisatmost Proof.Let(orderedbytheorderinginducedby).Thesetisamaximumfreesetinonlyif(a)isfreeand(b)foreachvertexwhichisnotin,theorderedset,h,risnotfree.Heretheissuchthatappearsbeforeintheorderingof,andappearsafterisorderedaccordingtotheorderingofTheprobabilitythatisfreehasbeenanalyzedinLemma5.13.Itislefttoanalyzetheprobabilitythatisnotfreeforeveryvertex,giventhatisfree.Consideravertexwhichisnotin,andlet,h,rClaim5.15.Letbeafreesetand,h,rLetthepartitioncorrespondingto,HI.Ifisalsoafreeset.Proof.Wewillusethefollowingnotation.Letdenotetheverticesof,andlet(,HI)beit
20 scorrespondingpartition.For,h,rdenotethe
scorrespondingpartition.For,h,rdenotetherst+1verticesof,andlet,HI)beitscorrespondingpartition.Finally,letdenotethesubsetand(,HI)beitscorrespondingpartition.Wewouldliketoprovethatisfree.Thatis,wewouldliketoshow(a)thatforeach,(b)that,(c)that,and(d)that+2.Recallthatisfree,andthusforallTherstassertionfollowsfromthefactthattherstverticesofidentical.Thesecondfollowsfromtheassumptionthat.Forthethirdas-sumption,notice(as)thatthepartitioncorrespondingtoisequaltothepartitioncorrespondingto.Thisfollowsfromourdenitionofthepartition(,HI).As,weconcludethatForthenalassertion,observethatforany+1,thepartitioncorrespondingisequaltothepartitioncorrespondingto.Thiscanbeseenbyinduction).Westartwiththepartitionscorrespondingto.Thepartition,HI)isdeneduniquelybythepartitioncorrespondingtoandthevertex.Similarly,thepartition(,HI)isdeneduniquelybythepartitioncorrespondingtoandthevertex.Asthepartitioncorrespondingisequaltothepartitioncorrespondingto,weconcludethatthesameholdforthepartiti
21 onscorrespondingto.Theinductivestepisdon
onscorrespondingto.Theinductivestepisdonesimilarly.Thepartitioncorrespondingto)isdeneduniquelybythepartitioncorrespondingto)andthevertex.Asthepartitioncorrespondingtoequalsthatcorrespondingto,weconcludeourclaim.Asisfree,forevery+2.Thisimpliesalsothat,whichprovesthenalasser- Claim5.15impliesthattheprobabilitythat,h,rnotfree,giventhatisfree,isatmost(1)(recallthatthesetisofsize).Thisholdsindependentlyforeveryvertex.Weconcludethat GRAPHSWITHTINYVECTORCHROMATICNUMBERSsucientlylargeconstant.If 2log thentheprobabilitythathasanindependentsetofsize�sisatmostProof.ByLemma5.17,isalso 42), -connected(pairwiseornot). )and .Wewouldliketoboundtheprobabilitythatdoesnothaveanyindependentsetsofsizegreaterthan.Let=1+ .Notice.Hence,itsucestoboundtheprobabilitythat�.Thisprobability,inturn,isatmosttheprobabilitythathasamaximumfreesetofsizegreaterthan(Claim5.11).Let=ln( .Itisnothardtoverifythat= )forourvalueof.Byourassumption,isgreaterthanorequalto 2log c1
22 t (1)2 log1 +log (1)2 c2t lo
t (1)2 log1 +log (1)2 c2t log1 +log whereintheabove,areconstantscloselyrelatedto.Now,byCorol-lary5.16,forourchoiceof,theprobabilitythathasamaximumfreesetofsizegreaterthanisatmost e(s)e(t). Roughlyspeaking,Theorem5.4statesthat,givena,-connectedgrapharandomsampleofsizeproportionalto (orlarger)willnothaveanindependentsetofsize(withhighprobability).Thisimprovesuponthebound presentedin[GGR98]bothinthedependenceon1)andinthedependenceon.Moreover,wepresentafurtherimprovementto ifourgraphsareconsideredtobepairwise,-connected.Insection6wecontinuetostudytheminimalvalueofforwhichswithhighprobability,andpresentalowerboundonthesizeofwhichisproportionalto 6.LowerboundsforthetestingofInthissectionwepresentgraphswhichare,-connected,butwithsomeconstantprobabilityarandomsampleofsizeislikelytohaveanindependentsetofsizegreaterthanLemma6.1.Letbeasmallconstantand.Forlargeenough,thereexistsagraphverticesforwhich,connected,andwithconstantprobability(independentof)a
23 randomsetofsize willhaveanindependentset
randomsetofsize willhaveanindependentsetofsizeProof.ConsiderthegraphV,E)inwhich,andconsistsoftwodisjointsets,whereisanindependentsetofsize(1 inducesaclique,andeveryvertexinisadjacenttoeveryvertexin.Ononehand,everysubsetofsizeinducesasubgraphwithatleast2edges(implyingthat,-connected).Ontheotherhand,letbearandomsubsetofobtainedbypickingeachvertexindependentlywithprobability .Theexpectedsizeof .Inthefollowingweassumethatisexactlyofsize;minormodicationsintheproofareneededifthisassumptionisnotmade.Thesetisanindependentsetinthesubgraphinducedby.Theexpectedsizeof .Let1)denoteastandardnormalvariable.Itcanbeseenusingthecentrallimittheorem(forexample,[Fel66])that,forourchoiceofparameters,theprobabilitythatdeviatesfromitsexpectationbymorethanasquarerootof GRAPHSWITHTINYVECTORCHROMATICNUMBERSTheorem7.2.Leta,k,beasdenedinTheorem.Thegraphispairwise-connected.Proof.Letbesubsets(in)ofsize.Thecorrespondingsubsetsarealsoofmeasure)(Lemma7.1).ByCorollary3.10,).WeconcludethatA,B Lem
24 ma7.3.Thegraphisvector -colorableforsome
ma7.3.Thegraphisvector -colorableforsomeconstantProof.Recallthateachcellinhasdiameteratmost2.Hence,twoverticesareconnectedonlyiftheirinnerproductislessthan1)+ k(1+(k) 2d)1. Bydenition,thecontinuousgraphisvector-colorable.InLemma7.3weshowedthattheniteapproximationvector-colorable.Ingeneral,thisdoesnotsucefortheproofofTheorem1.2,asweareinterestedingraphswhicharevector-colorable(ratherthanalmostvector-colorable).Thiscanbexedbystartingwithacontinuousgraphwithvectorcoloringnumberslightlylessthan ).Inordertosimplifyourpresentation,weignorethispointandconsiderthegraphtobeexactlyvector-colorable.Thisispossibleduetothefactthatthepropertiesofcontinuous.Namely,choosinglargeenough,itcanbeseenthatthemultiplicativeerrorof(1+ )inthevalueofdoesnotaecttheanalysisappearingthroughoutthissection.Lemma7.4.Let bethemeasureofa -cap.Everyvertexinthegraphhasdegree poly n,poly Proof.Consideravertexanditscorrespondingcell.Thedegreeofisthenumberofcellsinthatshareapositivemeasureofedges
25 withthecellThetotalmeasureofthesecellsis
withthecellThetotalmeasureofthesecellsisatleastthemeasureofa( ))-capandatmostthemeasureofa( ))-cap.Hence,byLemma7.1,weconcludeour WenowprovetherstpartofTheorem1.2byconsideringthegraphbyrandomlysamplingtheedgesofTheorem7.5.Foreveryconstantandconstant,thereareinnitelymanygraphsthatarevector-colorableandsatisfy ,whereisthenumberofverticesinisthemaximumdegreeinProof.Let2beconstant.Let0beanarbitrarilysmallconstant.Let /cforasucientlylargeconstant.LetV,E)bethediscretegraphdenedabove.Letbethesizeofthevertexset,andletbethemaximumdegreeof.Recallthat,whereisthedimensioninwhichthecorrespondinggraphwasdened.Wewillassumethatthedimensionisaverylargeconstantdeterminedafterxing.Finally,letByLemma7.4,allverticesinareofdegreeintherange[ poly],wherepoly .ByTheorem7.2andtheproofofClaim4.1,everysubsetofverticesofsizehasatleast k2+nedges.Letp(k ).Letbethesubgraphofobtainedbydeletingeachedgeofindependentlywithprobability(1Lemma7.6.Withprobability,allverticeswillhavedegreeinthe
26 range poly k22,2k k22. GRAPHSWI
range poly k22,2k k22. GRAPHSWITHTINYVECTORCHROMATICNUMBERSsubgraphsofhavebeenusedinthepastinthecontextunderdiscussion(see,e.g.,[KMS98,Fei97,GK98,Cha02]).Usingourprooftechniqueonsuchgraphsinvolvestheanalysisofcertainedgeisoperimetricinequalities(analogoustothosepresentedinTheorem3.5).Unfortunately,littleisknownregardingtheedgeisoperimetricinequalitiesoftheabovegraphs.Suchinequalitieshavebeenstudiedinthepast[Bez02,KKL88],yieldingpartialresults.However,theseresultsdonotsucetoextendourprooftechniques.Abetterunderstandingofedgeisoperimetricinequalitiesofthesegraphsisofgreatinterest,regardlessoftheirapplicationtothevectorcoloringPropertytesting.Insections5and6westudythepropertytestingparadigmwithrespecttotheindependentsetproblem.Wepresentimprovedresultsonthesamplesizeneededwhentestinggraphswhicharefromhavinglargeindependentsets.Namely,inTheorem5.4weprovethatifagraphofsize,thenwithhighprobabilityarandominducedsubgraphofofsizewillnothaveanindependentsetofsize.(T
27 hisimprovesuponthesamplesizeofpresentedi
hisimprovesuponthesamplesizeofpresentedin[GGR98].)Moreover,inLemma6.1weshowthatthesamplesizemustbeofsizeatleastifwewishtheprobabilityoffailuretobenonconstant.Itwouldbeinterestingifthefactorgapbetweentheupperandlowerboundspresentedabovecouldbesettled.9.Proofofclaims.Claim9.1.For,wehavethat Proof.Itiswellknownthatforanyreal.Hence, +p2y2 1pyep(y+py2 1py)(1p)y+py2 Asfortheotherdirection,forevery dp((1p)y(1(1p)y1)0. Claim9.2.Forall x1 e11 xx1 Proof.Itisknownthatforall xx1 e11 xx1. Claim9.3.Let�a�suchthat2log(;then Proof.Recallthat d1a2d1 2(a)1a2d1 ,forsomeconstantThus, 2(1a2)(12a)d1 21a2d1 22 d2(a) d. GRAPHSWITHTINYVECTORCHROMATICNUMBERS (a)(b)(c) x-x Fig.2Threecasesof,andarepresented.Thesetarepresentedbytheletter.Thesetarepresentedbytheletter.Apointin)ispresentedby.Apointinneitherisrepresentedbyasoliddot.Theedgesetisdepictedbysolidlines.Finally,thehyperplaneisrepresentedasahorizontalline.Eachcongurationi
28 spresentedbefore(above)andafter(below)th
spresentedbefore(above)andafter(below)thesymmetrizationprocedure.Incasethesetisclosedunderthesymmetrizationprocedure.Insuchcasesitholdsthat.Incasesnoticethatthereisadierencebetweenthevalueofandthenumberofundirectededgesbetweenthesets(forexample,incaseisofsizeConsiderthecaseinwhichconsistsoftheedges()and()only,thesubsetisequalto,andthesubsetisequalto.ThiscaseindepictedinFigure2(a).Inthisspeciccasewehavethatimplyingthat)=1.Noticethat,asarenitesets,wemeasuretheamountofedgesbetweenusingthediscreteanalogueof)denedinDenition5.1ofsection5(thesamegoesforThereare,ofcourse,severalothercasestoconsider(12edgecongurationsand256casesofdierentsubsets).SomeofthesecasesaredepictedinFigure2.Thislargecaseanalysismaybesignicantlyreducedusingvariousobservations(in-volvingtheequivalenceofmanydierentcases).Wehavecheckedourassertiononthefullcaseanalysis(usingacomputerprogram).Asimilarproofholdswhenarein(detailsomitted). Lemma9.5.isclosedinProof.Let()beasequenceintendingto(,)(inth
29 eHausdortopol-ogy).Wewillshowthat(,.F
eHausdortopol-ogy).Wewillshowthat(,.Forevery,�0wehaveforlargeenoughvaluestozeroandusingthefactthatforallclosedsets),weconcludethesecondpropertyof,namelythat)andFortherstproperty,sendingtozero,wehaveononehandthat).Fortheotherdirectionobservethat)and)forany0,providedthatislargeenough. GRAPHSWITHTINYVECTORCHROMATICNUMBERSUsingClaims9.2and9.3,weconclude(forsucientlylarge)that 1000z+2 10001 2z2 10001 31z2d1 beaconstantsuchthat �/10.Considerthesetz,u.Denotethe200neighborhoodof.Usingsphericalsymmetry,onecanupperboundthemeasureofthesetthemeasureofthecap ,whichisat z2+ 22 2002 d1 21z2d1 betheunionofthesets,/10+,/10+2 .Weconcludethatthemeasureofisatmost Finally,eachhz2/1000,z+2/1000]andand/10,/10]andthusisin.Hence\R ,implyingthatthemeasureisatleast 2µ(S)(1z2)d1 2. 3.7(restated)Leta,kbeasinTheorem.Let beavertexontheboundaryof.Letbethesetofneighborsof.Let a2+(1/(k1)+a2)2 .Finallylet log( forasucientlylargeconstant.The
30 measureofverticesin 2(1z2)d1 2µ(N(v))
measureofverticesin 2(1z2)d1 2µ(N(v))(1z2)d1 Proof.Thesetisequalto1)+ 1a2.Denote1/(k1)+a2 ).Let .Bysphericalsymmetry,itcanbeseenthattheabovesetisofmeasureatmost(1 ,andofmeasuregreaterorequaltothemeasureof=and0where1 )(foranyy,1/2],cisintherange(11)).Hence,itsucestoboundthemeasureofbybelow. log( forasucientlylargeconstant.Thesetaboveisofmeasurelargerthanthemeasureoftheneighborhoodof asabove.(TheneighborhoodofisdenedasinClaim9.7.)Wedenotethissetas.ByClaim9.7,themeasureofisatleastofvalue(1 (Noticethat,whichinturnisindependentof;thusClaim9.7canbeappliedinourcase.)Hence,themeasureofisatleast(1 2(1z2)d1 whichconcludesourproof. GRAPHSWITHTINYVECTORCHROMATICNUMBERS d+x d+xz2(1+c2c2z2) 1z2(1c2z2)andB= .Itisnothardtoverifythatforourvalueofitisthecasethatmin(.Thus 4Pr z2d 1z2r1 +1) and0 2wr1R1er21 2 AdAdr2=0r2er22 2=wr1R1er21 2 Ad(1eA2d 2)wr1R1Ader21 2w1x=0Adez2(d+x) 2(1z2)z 2 1z2 d+xdxwz2 d(1z2)(1c2z2)edz2 2(1z2)1x
31 =0xez2x 2(1z2)wz2 d(1z2)(1c2z2)edz
=0xez2x 2(1z2)wz2 d(1z2)(1c2z2)edz2 2(1z2)wz2 d(1z2)(1c2z2)1z2 1z2d1 2wz2 d(1z2)(1c2z2)1z2d1 212z4d1 istherange[ z2d 1z2, z2(d+1) ],andissomeconstantindependentof(thevalueofmaychangefromlinetoline).Aboveweusethefactthatandthefactthatisboundedbyabovebyaconstantindependentof.Inthenalinequalitiesweuseachangeofvariables z2(d+x) andthefactthat1+ 2for3.As poly (log(),weconcludeourassertion. FortheproofofTheorem3.8,let a2+(1/(k1)+a2)2 (asinClaim9.8);thus(log().Fortheupperbound,useaboundonthemeasureofandtheupperboundinClaim9.8.Asforthelowerbound,let=2(log( d)3 .Suchachoicewillsatisfylog( ,and(log(withrstcoordinateofvalue.ConsideravertexwithrstcoordinateofvaluelessthanByClaim9.8,itisnothardtoverifythatthemeasureofisgreaterthanthemeasureof,whichisgreaterthan polycz 2(1z2)d1 21 poly forsomeconstant.(WeuseClaim9.2andthefactthat(log().)As2log(,weconclude,usingCorollary9.3,thatthemeasureofedgesin)isatleast poly 2(a) poly 21 poly 21z2d1 2.