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Author Arora, Sanjeev ♦ Lund, Carsten ♦ Motwani, Rajeev ♦ Sudan, Madhu ♦ Szegedy, Mario
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1998
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword NP-completeness ♦ Optimization ♦ Proof verification ♦ Randomness
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 $\textit{constant}$ 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 [1998] 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 [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 et al. [1996] and Arora and Safra [1998] and show that there exists a positive ε such that approximating the maximum clique size in an $\textit{N}-vertex$ graph to within a factor of $\textit{N}ε$ is NP-hard.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1998-05-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 45
Issue Number 3
Page Count 55
Starting Page 501
Ending Page 555


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Source: ACM Digital Library