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1、THEPROBABILISTICMETHOD(b)Letb1,b2,...,bm/2allequal!,andleth111;2+J,bm/2+2,...,bmallequal-1.LetY1,Y2,...,Ymeachbechosenindependentlyanduniformlyatrandomfrom{0,1}.Showthatthereexistsapositiveconstantcsuchthat,forsufficientlylargem,Pr(ltb;Y;I>cJm)>�·l=l(Hint:Conditiononthenumberof}jthatareequalto1.)
2、(c)Letb1,b2,...,bmeachbeequaltoeither1or-1.LetY1,Y2,...,Ymeachbechosenindependentlyanduniformlyatrandomfrom{0,1}.Showthatthereexistsapositiveconstantcsuchthat,forsufficientlylargem,Pr(ltb;Y;I>cJm)>�·l=l(d)ProvethatthereexistsamatrixAforwhichIlAbII00isQ(fo)foranychoiceofb.Exercise6.16:UsetheLovasz
3、locallemmatoshowthat,if4(k)(n)21-(3)<1-'2k-2thenitispossibletocolortheedgesofKnwithtwocolorssothatithasnomonochromaticKksubgraph.Exercise6.17:UsethegeneralformoftheLovaszlocallemmatoprovethatthesymmetricversionofTheorem6.11canbeimprovedbyreplacingthecondition4dp::=:::1bytheweakerconditionep(d+1):
4、:=:::1.Exercise6.18:LetG=(V,E)beanundirectedgraphandsupposeeachvEVisassociatedwithasetS(v)of8rcolors,wherer:=::1.Suppose,inaddition,thatforeachvEVandcES(v)thereareatmostrneighborsuofvsuchthatcliesinS(u).ProvethatthereisapropercoloringofGassigningtoeachvertexvacolorfromitsclassS(v)suchthat,foranye
5、dge(u,v)EE,thecolorsassignedtouandvaredifferent.YoumaywanttoletAu.t·.cbetheeventthatuandvarebothcoloredwithcolorcandthenconsiderthefamilyofsuchevents.152CHAPTERSEVENMarkovChainsandRandomWalksMarkovchainsprovideasimplebutpowerfulframeworkformodelingrandomprocesses.Westartthischapterwiththebasicdef
6、initionsrelatedtoMarkovchainsandthenshowhowMarkovchainscanbeusedtoanalyzesimplerandomizedalgorithmsforthe2-SATand3-SATproblems.Nextwestudythelong-termbehadorofMarkovchains,explainingtheclassificationsofstatesandconditionsforconYergencetoastationarydistribution.Weapplythesetechniquestoanalyzingsim
7、plegamblingschemesandadiscreteversionofaMarkovianqueue.Ofspecialinterestisthelimitingbehaviorofrandomwalksongraphs.Weproveboundsonthecoveringtimeofagraphandusethisboundtodevelopasimplerandomizedalgorithmforthes-tconnec
