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Unformatted text preview: A Threshold of ln( n ) for approximating set cover October 20, 2009 1 The k-prover proof system • There is a single verifier V and k provers P 1 ,P 2 ,...,P k . • Binary code with k codewords, each of length l and weight l 2 , with Hamming distance at least l 3 . • Each prover is associated with a codeword. • The Protocol: – The verifier selects l clauses C 1 ,...,C l , then selects a variable from each clause to form a set of distinguished variables x 1 ,...x l . – Prover P i receives C j for those coordinates in its codeword that are 1 , and x j for the coordinates that are , and replies with 2 l bits. – The answer of the prover induces an assignment to the distinguished vari- ables. – Acceptance predicate: * Weak: at least one pair of provers is consistent. * Strong: every pair of provers is consistent. P 1 : 0011 P 2 : 0101 P 3 : 1100 V c 1 c 2 v 3 v 4 100,010,1,0 Figure 1: A k-prover proof system 1 L=4 k = 4 L=4 Figure 2: The partitioning system Lemma 1 Consider the k-prover proof system defined above and a 3CNF-5 formula φ . If φ is satisfiable, then the provers have a strategy that causes the verifier to always strongly accept. If at most a (1- )-fraction of the clauses in φ are simultaneously satisfiable, then the verifier weakly accepts with probability at most k 2 . 2- cl , where c > is a constant that depends on . Proof: • If φ is satisfiable, then provers can base their answers on a satisfying assignment. • Assume (1- ) clauses are satisfiable, and V weakly accepts with probability equal to δ , then with respect to P i and P j , V accepts with probability δ k 2 . • There are more than l/ 6 coordinates on which P i receives a clause, and P j re- ceives a variable in that clause. Fix the question pairs in the other 5 l/ 6 coordi- nates in a way that maximizes the acceptance probability, which by averaging remains at least δ/k 2 . • The provers have a strategy that succeeds with prob. = δ/k 2 on l/ 6 parallel repetitions of the original two-proof system. • From the Theorem 1 stated below from the reference  it follows that δ k 2 < 2- cl , hence, δ ≤ k 2 . 2- cl . Theorem 1 If a one-round two-prover proof system is repeated l times independently in parallel, then the error is 2- cl , where c > is a constant that depends only on the error of the original proof system (assuming this error was less than one) and on the length of the answers of the provers in the original proof system....
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