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Unformatted text preview: PCPs and Inapproximability The PCP Theorem and GAPMAXE3SAT My T. Thai 1 1 Recap 1.1 NPCompleteness Language L is in NP iff there is a polynomial time deterministic verifier V (say a Turning machine) and an arbitrarily powerful prover P , which the following properties: Completeness: For every x L , P can write a proof of length poly(  x  ) that V accepts Soundness: For every x / L , no matter what proof P writes, V rejects. For example, consider the Vertex Cover (VC) problem. Input x = ( G, k ) where G is a graph and k is a positive integer number. (Note that here is the decision problem of VC). The question here is that if there exists a VC of G with at most k vertices. The prover P will write a proof which is a subset S of vertices. The verifier C can verify if is a VC or not. Thus VC is in NP. 1.2 PCP The first question arising from the above definition is: What if we change the requirement of verifier V ? What if V produces onesidederror? That is, if x is a NO instance, V still output YES. What if V can read a certain amount of , not the whole as above? Randomness complexity : The total bits of randomness (usually is O (log n )) that V can uses, called r . In the case of NP as above, the randomness complexity is 0. Query complexity : The total bits of that V can read, called q . that is, V uses r bits of randomness to choose q random locations in and reads the bits in these q randomly chosen locations. Such a verifier V is called ( r, q )restricted verifier. More formally: Definition 1 V is said an ( r, q )restricted verifier V if V is a randomized polytime algorithm with randomness complexity r and query complexity q . 2 (Note that r and q can be a function of the input size, not necessary a con stant.) Given 0 s c 1, PCP c,s [ r, q ] if there exists an ( r, q )restricted verifier satisfying the following conditions: Completeness: If x is a YESinstance, then there exists a proof such that Prob[ V ( x, ) = Y ES ] c Soundness: If x is a NOinstance, then for any , Prob[ V ( x, ) = Y ES ] s When c = 1, and s = 1 / 2, for simplicity, we write PCP[ r, q ] instead of PCP 1 , 1 / 2 [ r, q ] Now, can you show that NP = PCP 1 ,s [0 , poly ( n )] for all s < 1? Can you show that PCP [ O (log n ) , O (log n )] NP ? Note that when r = O (log n ), the the proofchecking can be derandomized, that is, V can be simulated by a polynomial time deterministic verifier that simulates the computation of V on each of the 2 r = n O (1) possible random inputs and then computes the probability that V ( x, ) accepts, then accepts if and only if this probability is one....
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This note was uploaded on 05/20/2011 for the course CIS 6930 taught by Professor Staff during the Spring '08 term at University of Florida.
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