 
Summary: Proof Verification and the Hardness of
Approximation Problems
Sanjeev Arora # Carsten Lund + Rajeev Motwani #
Madhu Sudan § Mario Szegedy ¶
Abstract
We show that every language in NP has a probablistic verifier that checks mem
bership 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 SNPhard problem has a polynomial
time approximation scheme, unless NP=P. The class MAX SNP was defined by Pa
padimitriou 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
