TR98-008 Authors: Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, Mario Szegedy

Publication: 29th January 1998 11:19

Downloads: 815

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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

<em> constant </em> 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 [FOCS 1992]

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 [JCSS 1991]

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,

Lovasz, Safra and Szegedy [JACM 1996], and Arora and Safra [FOCS 1992]

and show that there exists a positive $\epsilon$ such that

approximating the maximum clique size in an N-vertex

graph to within a factor of N^{\epsilon} is NP-hard.

(An extended abstract of this paper appeared in FOCS 1992.

This is the full version.)