Proof verification and hardness of approximation problems

  • Authors:
  • S. Arora;C. Lund;R. Motwani;M. Sudan; Mario Szegedy

  • Affiliations:
  • Div. of Comput. Sci., California Univ., Berkeley, CA, USA;-;-;-;-

  • Venue:
  • SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
  • Year:
  • 1992

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Abstract

The class PCP(f(n),g(n)) consists of all languages L for which there exists a polynomial-time probabilistic oracle machine that used O(f(n)) random bits, queries O(g(n)) bits of its oracle and behaves as follows: If x in L then there exists an oracle y such that the machine accepts for all random choices but if x not in L then for every oracle y the machine rejects with high probability. Arora and Safra (1992) characterized NP as PCP(log n, (loglogn)/sup O(1)/). The authors improve on their result by showing that NP=PCP(logn, 1). The result has the following consequences: (1) MAXSNP-hard problems (e.g. metric TSP, MAX-SAT, MAX-CUT) do not have polynomial time approximation schemes unless P=NP; and (2) for some epsilon 0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup epsilon / unless P=NP.