Journal of the Association for Computing Machinery, Vol. 43, No. 2, March 1996, pp. 268–292.

Interactive Proofs and the Hardness of Approximating Cliques

URIEL FEIGE

SHAFI GOLDWASSER

LASZLO LOVÁSZ

SHMUEL SAFRA

AND

MARIO SZEGEDY

Abstract. The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interactive proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph.

Categories and Subject Descriptors. F.1.3 [Computation by Abstract Devices]: Complexity Classes—reducibility and completeness; relations among complexity classes; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems; G.2.2 [Discrete Mathematics]: Graph Theory

General Terms. Algorithms, Theory

Additional Key words and Phrases. Hardness of approximation, independent set in a graph, multilinearity testing, np-completeness, probabilistically checkable proofs

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