Proof verification and hardness of approximation problems

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Proof verification and hardness of approximation problems (1992)

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by Sanjeev Arora , Carsten Lund , Rajeev Motwani , Madhu Sudan , Mario Szegedy

Venue:	IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
Citations:	796 - 39 self

BibTeX

@INPROCEEDINGS{Arora92proofverification,
    author = {Sanjeev Arora and Carsten Lund and Rajeev Motwani and Madhu Sudan and Mario Szegedy},
    title = {Proof verification and hardness of approximation problems},
    booktitle = {IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI},
    year = {1992},
    pages = {14--23},
    publisher = {}
}

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Abstract

We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an N-vertex graph to within a factor of N ɛ is NP-hard.

Keyphrases

approximation problem proof verification constant number maximum clique size lov sz provided proof random bit max snp-hard problem nonconstant number n-vertex graph maximum cut hard problem maximum satisfiability polynomial time approximation scheme steiner tree random string input length probablistic verifier clique hardness result class max snp recent result membership proof logarithmic number metric tsp vertex cover

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