Recap Graph Isomorphism Probabilistically Checkable Proofs

JJ Advanced Complexity Theory April Lecture Lecturer Dan Spielman Scribe Ronnie Misra Recap Last time we discussed the class of ArthurMerln games Recall that Denition L AMkn if a PTIME verier A and a polynomial pn st 2 x L a pnprover P st PrA P accepts 3 1 x L pnprovers P PrA P accepts 3 In this lecture we will show some applications of IP and discuss the class PCP Graph Isomorphism Although there are no known PTIME algorithms for graph isomorphism we can show that ISO is probably not NPcomplete P P Theorem If ISO is NPcomplete then 3 3 Pro of In the problem set we showed that MA AM proved as NP BP P BP NP P In fact the AM sp eedup theorem says that k (n) AMkn AM 2 AMk AM constants k MAM AM We also showed in the problem set that AM BP NP P NPp oly ISO is NPcomplete NISO is coNPcomplete coNP MAM coNP AM coNP NPp oly P P 3 3 Probabilistically checkable pro ofs We would like to describ e a tough verier that accepts a pro of that can b e sp ot checked P Denition A Rn Qn verier is a probabilistic PTIME oracle TM M that gets Rn random bits is limited to Qn bits from its oracle P Denition L PCPRn Qn if a Rn Qn verier V st � x accepts x L st PrV 1 � x L PrV x accepts 2 Here is the probabilistically checkable pro of PCP Note that we dont really think of as an oracle Instead we are using OTM notation just as a way to show that V gets access to random bits of Theorem NEXP PCPPolyn Polyn Even though pro ofs for languages in NEXP are exp onentially long randomized access allows such pro ofs to b e veried in PTIME Theorem NP PCPOlog n O As a consequence we can use PCP to show that some approximation problems are NPhard SAT approximation is NPhard SAT NP PCPOlog n O a Olog n O verier for SAT Assume V is nonadaptive or that it ips R Olog n coins then queries the pro of at Q O places determined only by these random bits If V is adaptive we can construct a nonadaptive version this version may use exp onentially more queries but this is still a constant R On input assume V gets random bits r f g Let q q denote the indices of the r1 rQ V reads Q bits of that V queries Thus if 1 2 N q q r1 r Q R construct a small SAT instance on inputs and some For each r f g r q q r1 rQ auxilliary bits Assume has at most S clauses Since Q O S O should b e satised r r have values that would cause V to accept when q q r1 rQ Let r R r f01g is p olynomial in the size of and has inputs as well as all the auxilliary bits q q r1 rQ also has some sp ecial prop erties R SAT SAT since V will accept for an y r f g 1 SAT P r V rejects 2 R r f01g 1 at most of the clauses are satisable from aux bits r 2 1 aux there is at least one unsatised clause in at least of the clauses r 2 1 at least a fraction of the clauses of are unsatisable 2S 1 the maximum fraction of satisable clauses of is approximating SAT 2S 1 within a factor of is NPhard where S is some constant dep endant on Q 4S .

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