Denition1.1(UniqueGame).AuniquegameconsistsofaconstraintgraphG=(V;E), - PDF document

Denition1.1(UniqueGame).AuniquegameconsistsofaconstraintgraphG=(V;E),asetofvariablesxu(forallverticesu)andasetofpermutationsuvon[k]=f1;:::;kg(foralledges(u;v)).Eachpermutationuvdenestheconstraintuv(xu)=xv.Thegoalistoassignavaluefromtheset[k]toeachvariablexusoastomaximizethenumberofsatisedconstraints.Asinthesettingoflinearequations,instancesofuniquegameswhereallconstraintsaresatis-ableareeasytohandle.Givenaninstancewhere1�"fractionofconstraintsaresatisable,theUniqueGamesConjecture(UGC)ofKhot[10]saysthatitishardtosatisfyevenfractionoftheconstraints.Moreformally,theconjectureisthefollowing.Conjecture1(UniqueGamesConjecture[10]).Foranyconstants";�0,foranyk�k(";),itisNP-hardtodistinguishbetweeninstancesofuniquegameswithdomainsizekwhere1�"fractionofconstraintsaresatisableandthosewherefractionofconstraintsaresatisable.Thisconjecturehasattractedalotofrecentattentionsinceithasbeenshowntoimplyhardnessofapproximationresultsforseveralimportantproblems:MaxCut[11,15],Min2CNFDeletion[3,10],MultiCutandSparsestCut[3,14],VertexCover[13],andcoloring3-colorablegraphs[5](basedonavariantoftheUGC),thatseemdiculttoobtainbystandardcomplexityassumptions.Notethatarandomassignmentsatisesa1=kfractionoftheconstraintsinauniquegame.Andersson,Engebretsen,andHastad[2]consideredsemideniteprogram(SDP)basedalgorithmsforsystemsoflinearequationsmodp(withtwovariablesperequation)andgaveanalgorithmthatperforms(veryslightly)betterthanarandomassignment.TherstapproximationalgorithmforgeneraluniquegameswasgivenbyKhot[10],andsatises1�O(k2"1=5p log(1="))fractionofallconstraintsif1�"fractionofallconstraintsissatisable.RecentlyTrevisan[17]developedanalgorithmthatsatises1�O(3p "logn)fractionofallconstraints(thiscanbeimprovedto1�O(p "logn)[9]),andGuptaandTalwar[9]developedanalgorithmthatsatises1�O("logn)fractionofallconstraints.Theresultof[9]isbasedonroundinganLPrelaxationfortheproblem,whilepreviousresultsuseSDPrelaxationsforuniquegames.Thereareveryfewresultsthatshowhardnessofuniquegames.FeigeandReichman[7]showedthatforeverypositive"thereiscs.t.itisNP-hardtodistinguishwhethercfractionofallconstraintsissatisable,oronly"cfractionissatisable.OurResults.Wepresenttwonewapproximationalgorithmsforuniquegames.Westateourguaranteesforinstanceswhere1�"fractionofconstraintsaresatisable.Therstalgorithmsatises min(1;1 p "logk)
(1�")2
k p logk�"=(2�")!(1)fractionofallconstraints.Thesecondalgorithmsatises1�O(p "logk)fractionofallconstraintsandhasabetterguaranteefor"=O(1=logk).Weapplythesametechniquesford-to-1gamesaswell.Inordertounderstandthecomplexitytheoreticimplicationsofourresults,itisusefultokeepinmindthatinapproximabilityreductionsfromuniquegamestypicallyusethe\LongCode",whichincreasesthesizeoftheinstancebya2kfactor.Thus,suchapplicationsofuniquegamesusuallyhavedomainsizek=O(logn).InFigure1,wesummarizeknownalgorithmicguaranteesforuniquegames.Inordertocomparethesedierentguaranteesinthecontextofhardnessapplications(i.e.k=O(logn)),wecomparetherangeofvaluesof"(asafunctionofk)forwhicheachofthesealgorithmsbeatstheperformanceofarandomassignment.2 everyvariableu,oneforeveryvaluei2[k].Givenaconstraintuvonuandv,thevectorsuiandvuv(i)areclose.IncontrasttothealgorithmsofTrevisan[17]andGupta,Talwar[9],ourroundingalgorithmsignoretheconstraintgraphentirely.WeinterprettheSDPsolutionasaprobabilitydistributiononassignmentsofvaluestovariablesandthegoalofourroundingalgorithmistopickanassignmenttovariablesbysamplingfromthisdistributionsuchthatvaluesofvariablesconnectedbyconstraintsarestronglycorrelated.Theroughideaistopickarandomvectorandexaminetheprojectionsofthisvectoronui,pickingavalueiforuforwhichuihasalargeprojection.(Infact,thisisexactlythealgorithmofKhot[10]).Wehavetomodifythisbasicideatoobtainourresultssincetheui'scouldhavedierentlengthsandothercomplicationsarise.Insteadofpickingonerandomvector,wepickseveralGaussianrandomvectors.Ourrstalgorithm(suitableforlarge")picksasmallsetofcandidateassignmentsforeachvariableandchoosesrandomlyamongstthem(independentlyforeveryvariable).Itisinterestingtonotethatsuchamultipleassignmentisoftenencounteredinalgorithmsimplicitinhardnessreductionsinvolvinglabelcover.Incontrasttopreviousresults,thisalgorithmhasanon-trivialguaranteeevenforverylarge".As"approaches1(i.e.forinstanceswheretheoptimalsolutionsatisesonlyasmallfractionoftheconstraints),theperformanceguaranteeapproachesthatofarandomassignment.Oursecondalgorithm(suitableforsmall")carefullypicksasingleassignmentsothatalmostallconstraintsaresatised.TheperformanceguaranteeofthisalgorithmgeneralizesthatobtainedbyGoemansandWilliamson[8]fork=2.Notethatauniquegameofdomainsizek=2where1�"fractionofconstraintsissatisableisequivalenttoaninstanceofMax-Cutwheretheoptimalsolutioncuts1�"fractionofalledges.Forsuchinstances,therandomhyperplaneroundingalgorithmof[8]givesasolutionofvalue1�O(p "),andourguaranteecanbeviewedasageneralizationofthistolargerk.Thereadermightwonderaboutthecon uenceofourboundsandthelowerboundsobtainedbyKhotetal.[12].Infact,botharisefromtheanalysisofthesamequantity:Giventwounitvectorswithdotproduct1�",conditionedontheprobabilitythatonehasprojection(p logk)onarandomGaussianvector,whatistheprobabilitythattheotherhasalargeprojectionaswell?Thisquestionarisesnaturallyintheanalysisofourroundingalgorithms.Ontheotherhand,theboundsobtainedbyKhotetal.[12]dependonthenoisestabilityofcertainfunctions.Viatheresultsof[15],thisisboundedbytheanswertotheabovequestion.InSection2,wedescribethesemideniterelaxationforuniquegames.InSections3and4,wepresentandanalyzeourapproximationalgorithms.InSection5,weapplyourresultstod-to-1games.WedefersomeofthetechnicaldetailsofouranalysistotheAppendix.Recently,Chlamtac,MakarychevandMakarychev[4]havecombinedourapproachwithtech-niquesofmetricembeddings.Theirapproximationalgorithmforuniquegamessatises1�O("p lognlogk)fractionofallconstraints.ThisgeneralizestheresultofAgarwal,Charikar,Maka-rychev,andMakarychev[1]fortheMinUnCutProblem(i.e.thecasek=2).Notethattheirapproximationguaranteeisnotcomparablewithours.2SemideniteRelaxationFirstwereduceauniquegametoanintegerprogram.Foreachvertexuweintroducekindicatorvariablesui2f0;1g(i2[k])fortheeventsxu=i.Foreveryu,theintendedsolutionhasui=1forexactlyonei.Theconstraintuv(xu)=xvcanberestatedinthefollowingform:foralliui=vuv(i):4 productwithgareabovesomethresholdtothesetSu;wechoosethethresholdinsuchawaythatthesetSucontainsonlyoneelementinexpectation.ThenpickarandomstatefromSuandassignittothevertexu(ifSuisemptydonotassignanystatestou).Whatistheprobabilitythatthealgorithmsatisesaconstraintbetweenverticesuandv?Looselyspeaking,thisprobabilityisequaltoEjSu\uv(Sv)j jSuj
jSvjE[jSu\uv(Sv)j]:AssumeforamomentthattheSDPsolutionissymmetric:thelengthsofallvectorsuiarethesameandthesquaredEuclideandistancebetweeneveryuiandvuv(i)isequalto2".(Infact,theseconstraintscanbeaddedtotheSDPinthespecialcaseofsystemsoflinearequationsoftheformxi�xj=cij(modp).)SincewewanttheexpectedsizeofSutobe1,wepickthresholdsuchthattheprobabilitythathg;uiiequals1=k.Therandomvariableshg;p k
uiiandhg;p k
vuv(i)iarestandardnormalrandomvariableswithcovariance1�"(notethatwemultipliedtheinnerproductsbyanormalizationfactorofp k).Forsuchrandomvariablesiftheprobabilityoftheeventhg;p k
uiitp kequals1=k,thenroughlyspeakingtheprobabilityoftheeventhg;p k
uiitp k
andhg;p k
vuv(i)itp k
equalsk�"=2
1=k.ThustheexpectedsizeoftheintersectionofthesetsSuanduv(Sv)isapproximatelyk�"=2.Unfortunatelythisnolongerworksifthelengthsofvectorsaredierent.Themainproblemisthatif,say,u1istwotimeslongerthanu2,thenPr(u12Su)ismuchlargerthanPr(u22Su).Oneofthepossiblesolutionsistonormalizeallvectorsrst.InordertotakeintoaccountoriginallengthsofvectorswerepeattheprocedureofaddingvectorstothesetsSumanytimes,buteachvectoruihasachancetobeselectedinthesetSuonlyintherstsu;itrials,wheresu;iissomeintegernumberproportionaltotheoriginalsquaredEuclideanlengthofui.WenowformallypresentaroundingalgorithmfortheSDPdescribedintheprevioussection.InAppendixD,wedescribeanalternateapproachtoroundingtheSDP.Theorem3.1.Thereisapolynomialtimealgorithmthatndsanassignmentofvariableswhichsatises min(1;1 p "logk)
(1�")2
k p logk�"=(2�")!fractionofallconstraintsiftheoptimalsolutionsatises(1�")fractionofallconstraints.RoundingAlgorithm1Input:AsolutionoftheSDP,withtheobjectivevalue"
jEj.Output:Anassignmentofvariablesxu.1.Dene~ui=ui=juijifui6=0,0otherwise.Notethatvectors~u1;:::;~ukareorthogonalunitvectors(exceptforthosevectorsthatareequaltozero).2.PickrandomindependentGaussianvectorsg1;:::;gkwithindependentcomponentsdis-tributedasN(0;1).3.Foreachvertexu:(a)Setsui=djuij2
ke.6 Denition3.4.Forbrevity,denote�p logk=k
2=(2�x)byfk(x).Remark3.1.ItisinstructivetoconsiderthecasewhentheSDPsolutionisuniforminthefol-lowingsense:1.Thelengthsofallvectorsuiarethesameandareequalto1=p k.2.All"iuvareequalto".Inthiscaseallsuiareequalto1.Andthustheprobabilitythataconstraintissatisedisktimestheprobability(6)whichisequal,uptoalogarithmicfactor,tok�"=(2�").Multiplyingthisprobabilitybythenumberofedgeswegetthattheexpectednumberofsatisedconstraintsisk�"=(2�")jEj.Inthegeneralcasehoweverweneedtodosomeextraworktoaveragetheprobabilitiesamongallstatesiandedges(u;v).Recallthatweinterpretjuij2astheprobabilitythatthevertexuisinthestatei.Supposenowthattheconstraintbetweenuandvissatised,whatistheconditionalprobabilitythatuisinthestateiandvisinthestateuv(i)?Roughlyspeaking,itshouldbeequalto(juij2+jvuv(i)j2)=2.Thismotivatesthefollowingdenition.Denition3.5.Deneameasureuvontheset[k]:uv(T)=Xi2Tjuij2+jvuv(i)j2 2;whereT[k]:Notethatuv([k])=1.Thisfollowsfromconstraint(3).Thefollowinglemmashowswhythismeasureisuseful.Lemma3.6.Foreveryedge(u;v)thefollowingstatementshold.1.Theaveragevalueof"iuvw.r.t.themeasureuvislessthanorequalto"uv:kXi=1uv(i)"iuv"uv:2.Foreveryi,min(sui;svuv(i))(1�"iuv)2uv(i)k:Proof.1.Indeed,kXi=1uv(i)
"iuv=kXi=1juij2+jvuv(i)j2�(juij2+jvuv(i)j2)
cosi 2kXi=1juij2+jvuv(i)j2�2
juij
jvuv(i)j
cosi 2=kXi=1jui�vuv(i)j2 2="uv8 Proof.LetusrestrictourattentiontoasubsetofedgesE0=f(u;v)2E:"uv2"g.For(u;v)inE0,since"uvlogk2"logk,wehavePuv= k p logkmin(1;1 p "logk)(1�"uv)2
fk("uv):Summingthisprobabilityoveralledges(u;v)inE0andusingconvexityofthefunction(1�x)2fk(x)wegetthestatementofthetheorem. 4AlmostSatisableInstancesSupposethat"isO(1=logk).Intheprevioussectionwesawthatinthiscasethealgorithmndsanassignmentofvariablessatisfyingaconstantfractionofconstraints.Butcanwedobetter?Inthissectionweshowhowtondanassignmentsatisfying1�O(p "logk)fractionofconstraints.ThemainissueweneedtotakecareofistoguaranteethatthealgorithmalwayspicksonlyoneelementinthesetSu(otherwisewelooseaconstantfactor).Thiscanbedonebyselectingthelargestinabsolutevalueui;s(atstep3.d).Wewillalsochangethewaywesetsui.Denoteby[x]rthefunctionthatroundsxupordowndependingonwhetherthefractionalpartofxisgreaterorlessthanr.Notethatifrisarandomvariableuniformlydistributedintheinterval[0;1],thentheexpectedvalueof[x]risequaltox.RoundingAlgorithm2Input:AsolutionoftheSDP,withtheobjectivevalue"
jEj.Output:Anassignmentofvariablesxu.1.Pickanumberrintheinterval[0;1]uniformlyatrandom.2.PickrandomindependentGaussianvectorsg1;:::;g2kwithindependentcomponentsdis-tributedasN(0;1).3.Foreachvertexu:(a)Setsui=[2k
juij2]r.(b)Foreachiprojectsuivectorsg1;:::;gsuito~ui:ui;s=hgs;~uii;1ssui:(c)Selectui;swiththelargestabsolutevalue,wherei2[k]andssui.Assignxu=i.WerstelaborateonthedierencebetweenthechoiceofsuiinthealgorithmaboveandthatinAlgorithm1presentedearlier.Consideraconstraintuv(xu)=xv.Projectionui;sgeneratedbyuiandvuv(i);sgeneratedbyvuv(i)areconsideredtobematched.Ontheotherhand,aprojectionui;ssuchthatthecorrespondingvuv(i);sdoesnotexist(orviceversa)isconsideredtobeunmatched.Unmatchedprojectionsarisewhensui6=svuv(i)andthefractionofsuchprojectionslimitstheprobabilityofsatisfyingtheconstraint.RecallthatinAlgorithm1,wesetsui=djuij2
ke.Evenifuiandvuv(i)areinnitesimallyclose,itmayturnoutthatsuiandsvuv(i)dierby1,yieldinganunmatchedprojection.Asaresult,someconstraintsthatarealmostsatisedbytheSDPsolution(i.e."uviscloseto0)couldbesatisedwithlowprobability(bytherstroundingalgorithm).In10 Lemma4.3.1.TheexpectedsizeofMcisatmost4"uvk:E[jMcj]4"uvk:2.ThesetMalwayscontainsatleastk=2elements:jMjk=2.Proof.1.Firstwendtheexpectedvalueofjsui�svuv(i)jforaxedi.ThisvalueisequaltoEr[2k
juij2]r�[2k
jvuv(i)j2]r=2k
juij2�jvuv(i)j2:Nowbythetriangleinequalityconstraint(5),2k
juij2�jvuv(i)j22k
jui�vuv(i)j2:Summingoveralliin[k]wenishtheproof.2.Observethatmin(sui;svuv(i))2k
min(juij2;jvuv(i)j2)�1andmin(juij2;jvuv(i)j2)juij2�jjuij2�jvuv(i)j2jjuij2�jui�vuv(i)j2:SummingoveralliwegetjMj=Xi2[k]min(sui;svuv(i))Xi2[k]�2k
juij2�2k
jui�vuv(i)j2�1
2k�4k"uv�kk=2: Lemma4.4.Thefollowinginequalityholds:E241 jMjX(i;s)2M"iuv354"uv:Proof.RecallthatMalwayscontainsatleastk=2elements.Theexpectedvalueofmin(sui;svuv(i))isequalto2k
min(juij2;jvuv(i)j2)andislessthanorequalto2k
uv(i).ThuswehaveEr241 jMjX(i;s)2M"iuv35=Er"1 jMjkXi=1min(sui;svuv(i))
"iuv#2 kkXi=12k
uv(i)
"iuv4kXi=1uv(i)
"iuv4"uv: 12 Notethatevenifallconstraintsofad-to-1gamearesatisableitishardtondanassignmentofvariablessatisfyingallconstraints.Wewillshowhowtosatisfy 0@1 p logk
(1�")4
k p logk�p d�1+" p d+1�"1Afractionofallconstraints(themultiplicativeconstantinthe notationdependsond).Noticethatthisvaluecanbeobtainedbyreplacing"informula(1)with"0=1�(1�")=p d(andchanging(1�")2to(1�")4).Eventhoughwedonotrequirethatforaconstraintuveachiin[k]belongstosomepair(i;j)2uv,letusassumethatforeachithereexistsjs.t.(i;j)2uv;andforeachjthereexistsis.t.(i;j)2uv.Asweseelaterthisassumptionisnotimportant.Inordertowritearelaxationford-to-1gamesintroducethefollowingnotation:wiuv=Xj:(i;j)2uvvj:TheSDPisasfollows:minimize1 2X(u;v)2E kXi=1ui�wiuv2!subjectto8u2V8i;j2[k];i6=jhui;uji=0(9)8u2VkXi=1juij2=1(10)8(u;v)2Vi;j2[k]hui;vji0(11)8(u;v)2Vi2[k]0hui;wiuvimin(juij2;jwiuvj2)(12)Animportantobservationisthatjw1uvj2+:::+jwkuvj2=1,hereweusethefactthatforaxededge(u;v)eachvjisasummandinoneandonlyonewiuv.WeuseAlgorithm1forroundingavectorsolution.Foranalysiswewillneedtochangesomenotation:~wiuv=(wiuv=jwiuvj;ifwiuv6=0;0;otherwise"iuv=j~ui�~wiuvj2 2"iuv0=1�1�"iuv p duv(i)=juij2+jwiuvj2 2Thefollowinglemmaexplainswhywegetthenewdependencyon".14 Wenowaddresstheissuethatforsomeedges(u;v)andstatesjtheremaynotnecessarilyexistis.t.(i;j)2uv.Wecallsuchjastateofdegree0.Thekeyobservationisthatinouralgorithmswemayenforceadditionalconstraintslikexu=iorxu6=ibysettingui=1orui=0respectfully.Thuswecanaddextrastatesandenforcethattheverticesarenotinthesestates.Thenweaddpairs(i;j)whereiisanewstate,andjisastateofdegree0(orvice-versa).Alternativelywecanrewritetheobjectivefunctionbyaddinganextraterm:minimize1 2X(u;v)2E kXi=1ui�wiuv2+jw0uvj2!;wherew0uvisthesumofvjoverjofdegree0.AcknowledgementsWewouldliketothankSanjeevArora,UriFeige,JohanHastad,MuliSafraandLucaTrevisanforvaluablediscussionsandEdenChlamtacforhiscomments.ThesecondandthirdauthorsthankMicrosoftResearchfortheirhospitality.References[1]A.Agarwal,M.Charikar,K.Makarychev,andY.Makarychev.O(p logn)approximationalgorithmsforMinUnCut,Min2CNFDeletion,anddirectedcutproblems.InProceedingsofthe37thAnnualACMSymposiumonTheoryofComputing,pp.573{581,2005.[2]G.Andersson,L.Engebretsen,andJ.Hastad.Anewwaytousesemideniteprogrammingwithapplicationstolinearequationsmodp.JournalofAlgorithms,Vol.39,pp.162{204,2001.[3]S.Chawla,R.Krauthgamer,R.Kumar,Y.Rabani,andD.Sivakumar.Onthehardnessofapproximatingmulticutandsparsest-cut.InProceedingsofthe20thIEEEConferenceonComputationalComplexity,pp.144{153,2005.[4]E.Chlamtac,K.MakarychevandY.Makarychev.HowtoplayanyUniqueGame.manuscript,February2006.[5]I.Dinur,E.Mossel,andO.Regev.Conditionalhardnessforapproximatecoloring.ECCCTechnicalReportTR05-039,2005.[6]U.FeigeandL.Lovasz.Two-proveroneroundproofsystems:Theirpowerandtheirproblems.InProceedingsofthe24thACMSymposiumonTheoryofComputing,pp.733{741,1992.[7]U.FeigeandD.Reichman.Onsystemsoflinearequationswithtwovariablesperequa-tion.InProceedingsofthe7thInternationalWorkshoponApproximationAlgorithmsforCombinatorialOptimization,vol.3122ofLectureNotesinComputerScience,pp.117{127,2004.16 2.Thereexistpositiveconstantsc1;C1;c2;C2suchthatforall0p1=3,t0and1thefollowinginequalitieshold:c1 p 2(t+1)e�t2 2~(t)C1 p 2(t+1)e�t2 2;c2p log(1=p)~�1(p)C2p log(1=p);3.ThereexistsapositiveconstantC3,s.t.forevery02andt1=thefollowinginequalityholds:~(t+1 t)C3(t
~(t))2
t�1:Proof.1.Firstnoticethat~(t)=1 p 2Z1te�x2 2dx=1 p 224�e�x2 2 x1t�Z1te�x2 2 x2dx35=1 p 2te�t2 2�1 p 2Z1te�x2 2 x2dx:Thus~(t)1 p 2te�t2 2:Ontheotherhand1 p 2Z1te�x2 2 x2dx1 p 2t2Z1te�x2 2dx=~(t) t2:Hence~(t)�1 p 2te�t2 2�~(t) t2;and~(t)�t p 2(t2+1)e�t2 2:2.Thistriviallyfollowsfrom(1).3.Using(2)weget~(t+1 t)C
1+t+1 t�1
e�(t+1 t)2 2C0
(t+1)�1
e�2
t2 2C000@e�t2 2 t+11A2
t2
t�1C000
(t
~(t))2
t�1 18 Noticethat1=221.Wenowestimatetheprobability(13)asfollowsPr�tandt
=PrXt+p 1�2
jYjPrXt + t
PrjYj2 p 1�2
tByLemmaA.1(3)wegetPr�tandt
C
t�1
(t~(t))1=2
min1;2 p 1�2
tC0
min((p "
t)�1;1)
t�1
(t
~(t))2 2�": CorollaryB.2.Letandbestandardnormalrandomvariableswithcovariancegreaterthanorequalto1�";let~(t)=1=k.ThenPr(tandt) min1;1 p "logk
1 p logk
k p logk�2 2�"!:LemmaB.3.Let,,",kandtbeasinCorollaryB.2,andlet1;:::;mbei.i.d.standardnormalrandomvariablesandm2k,thenE"mXi=1Ifitgjtandt#=O(1);whereIfitgistheindicatoroftheeventfitg.Proof.LetXandYbeasintheproofofLemmaB.1.Puti=cov(X;i)andexpresseachiasi=iX+q 1�2i
Zi.ByBessel'sInequality21+


+2m1(sincerandomvariablesiareorthogonal).WenowestimatePr(itjXt+p 1�2
jYj)=PritjXt+p 1�2
jYjandX4t
PrX4tjXt+p 1�2
jYj+PritjXt+p 1�2
jYjandX�4t
PrX�4tjXt+p 1�2
jYjNoticethatPr�itjXt+p 1�2
jYjandX4t
=Pr�iX+q 1�2i
ZitjXt+p 1�2
jYjandX4t
=Z4tt=Pr�ix+q 1�2i
Zitjxt+p 1�2
jYj
dF(x)maxx2[t=;4t]Prq 1�2i
Zit�ixjp 1�2
jYjx�tbyCorollaryA.3maxx2[t=;4t]Prq 1�2i
Zit�ixPr(Zi(1�4i)t):20 Proof.WehavePrtand+t=Z10Pr(tand+xt)dF(x)=1 p 2Z10Pr(tand+xt)e�x2 22dx=1 p 2Z10Pr(tand+yt)e�y2 2dy=1 p 2Zt=0Pr(t�yt)e�y2 2dy+1 p 2Z1t=Pr(t�yt)e�y2 2dy:Letusboundtherstintegral.Sincethedensityoftherandomvariableontheinterval(t�y;t)isatmost1 p 2e�(t�y)2 2andyey,wehavePr(t�yt)y
1 p 2e�(t�y)2 2 p 2
e�t2 2
e(t+1)y:Therefore,1 p 2Zt=0Pr(t�yt)e�y2 2dye�t2 2 2Zt=0e(t+1)y
e�y2 2dye�t2 2 2Z1�1e�(y�(t+1))2 2
e(t+1)2 2dy=Oe�t2 2
e(t+1)2 2:Wenowupperboundthesecondintegral.Ift1,then1 p 2Z1t=Pr(t�yt)e�y2 2dy1 p 2Z1t=e�y2 2dy=~(t=)=O0@e�t2 22 t=+11A=O0@e�t2 2 t+1A=Oe�t2 2:Ift1,then1 p 2Z1t=Pr(t�yt)e�y2 2dy1 p 2Z10y
e�y2 2dy=O()=Oe�t2 2:Thedesiredinequalityfollowsfromtheupperboundsontherstandsecondintegrals. Weneedaslightgeneralizationofthelemma.CorollaryC.2.Letandbetwoindependentrandomnormalvariableswithvariance1and2respectively(01).Thenforeveryt0and0"1Pr(+(1�")tjjjt)=O (+"t)
c(";;t)
e�t2=2 1�2~(t)!;22 similarly,2=21+22+2:Notethat1=cov(1;1)1�"and1=cov(1;2)0.ThusVar[1]Var[1]�211�(1�")22":Similarly,20,21�",andVar[2]2".Since1and2areindependent,wehave12+12+cov(1;2)=cov(1;2)=0:Therefore(notethatcov(1;2)0;120;120),2=�12�cov(1;2) 1p Var[1]Var[2] 1�"2" 1�":Takingintoaccountthat21,weget2min(1;2" 1�")3".Similarly,13".Finally,wehavej1j�j2j(1�2)j1j�(1+2)j2j�j1j�j2j(1�4")j1j�(1+3")j2j�j1j�j2j: Inwhatfollowsweassumethat1isthelargestr.v.inabsolutevalueamong1;:::;manditsabsolutevalueist.Forconveniencewedenethreeevents:At=fjijtforall3img;Et=At\fj1j=tandj2jtg;E=fj1jjijforallig=[t0Et:NowwearereadytocombineCorollaryC.2andLemmaC.3.LemmaC.4.Let1;


;mand1;


;mbetwosequencesofstandardnormalrandomvariables.Supposethat1.therandomvariablesineachofthesequencesareindependent,2.thecovarianceofeveryiandjisnonnegative,3.cov(1;1)1�"andcov(2;2)1�",where"1=7.ThenPr(j1jj2jjEt)=O (p "+"t)
e�t2=2
c(7";p 8";t) 1�2~(t)!;(15)wherec(";;t)isfromCorollaryC.2.24 Noticethat(p "+"t)
e�t2=2 1�2~(t)=O (p "+"t)
(t+1)
~(t) 1�2~(t)!;since~(t)= e�t2=2 t!:2.If"�1=32thestatementholdstrivially.Soassumethat"1=32.Then(p 8"t+1)2 2+7"t23t2 8+O(t):Thust
e�t2 2
c(7";p 8";t)isupperboundedbysomeabsoluteconstant.Sincet1,thedenominator1�2~(t)oftheexpression(15)isboundedawayfrom0. Wenowgiveaboundonthe\typical"absolutevalueofthelargestrandomvariable.LemmaC.6.Thefollowinginequalityholds:Prj1j2p logmjE1 m:Proof.NotethattheprobabilityoftheeventEis1=m,sinceallrandomvariables1;:::;mareequallylikelytobethelargestinabsolutevalue.ThuswehavePrj1j2p logmjEPr�j1j2p logm
Pr(E)1 m21 m=1 m: LemmaC.7.Let1;


;mand1;


;mbetwosequencesofstandardnormalrandomvari-ablesasinTheorem4.1.Assumethatcov(1;1)1�"andcov(2;2)1�",where"min(1=(4logm);1=7).ThenPr(j1jj2jjE)=Op "logm m:Proof.Writethedesiredprobabilityasfollows:Pr(j1jj2jjE)=Prj1jj2jandj1j2p logmjE+Prj1jj2jandj1j2p logmjEFirstconsiderthecasej1j2p logm.DenotebydFj1jthedensityofj1jconditionalonE.ThenPrj1jj2jandj1j2p logmjE=Z2p logm0Pr(j1jj2jjEandj1j=t)dFj1j(t)=Z2p logm0Pr(j1jj2jjEt)dFj1j(t)26 Applyingtheunionbound,wegetPr(j1jmaxi2jjjjj1jmaxj2jjj)=O p logm mmXi=2q max("1;"i)!=O p logm m
mp "1+mXi=1p "i!!byJensen'sinequalityOp logm(p "1+p "):Sincetheprobabilitythatjij=maxjjjjequals1=mforeachi,theprobabilitythatthelargestr.v.inabsolutevalueamongi,andthelargestr.v.inabsolutevalueamongjhavedierentindexesisatmostO 1 mmXi=1p logm
(p "i+p ")!Op logm
(p "+p ")=Op "logm: ProofofTheorem4.1.Denote"i=1�cov(i;i).Then("1+


+"m)m".Wemayassumethat"min(1=(4logm);1=7)|otherwise,thetheoremfollowstrivially.ConsiderthesetI=fi:"imin(1=(4logm);1=7)g.Since"min(1=(4logm);1=7),thesetIisnotempty.ApplyingLemmaC.8torandomvariablesfigi2Iandfigi2I,weconcludethatthethelargestr.v.inabsolutevalueamongfigi2Ihasthesameindexasthelargestr.v.inabsolutevalueamongfigi2Iwithprobability1�O0@s logjIj
1 jIjXi2I"i1A=1�Op "logm:Sinceeachiisthelargestr.v.among1,...,minabsolutevaluewithprobability1=m,theprobabilitythatthelargestr.v.among1,...,mdoesnotbelongtofigi2Iism�jIj m.Similarly,theprobabilitythatthelargestr.v.among1,...,mdoesnotbelongtofigi2Iis(m�jIj)=m.Therefore,bytheunionbound,theprobabilitythatthelargestr.v.inabsolutevalueamongi,andthelargestr.v.inabsolutevalueamongjhavedierentindexesisatmost1�O(p "logm)�2m�jIj m:(16)Wenowupperboundthelastterm.2m�jIj mbytheMarkovinequality2" min(1=(4logm);1=7)2(4logm+7)"=O("logm)=O(p "logm):(Hereweusethat"logm1.)Pluggingthisboundinto(16)wegetthatthedesiredprobabilityis1�O(p "logm).Thisnishestheproof. 28