More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP

• Authors: Lars Engebretsen;Jonas Holmerin
• Venue: Random Structures & Algorithms
• Year: 2008

Samorodnitsky and Trevisan [STOC 2000, pp. 191199] proved thatthere exists, for every positive integer k, a PCP forNP with O(log n) randomness, query complexity2k + k2, free bit complexity 2k,completeness 1 - ε, and soundness2-k2 + ε. In this article, wedevise a new "outer verifier," based on the layered label coverproblem recently introduced by Dinur et al. [STOC 2003, pp.595601], and combine it with a new "inner verifier" that uses thequery bits more efficiently than earlier verifiers. Our resultingtheorem is that there exists, for every integer f ≥ 2,every positive integer t ≤ f(f - 1)-2, andevery constant ε 0, a PCP for NP withO(log n) randomness, query complexity f +t, free bit complexity f, completeness 1 -ε, and soundness 2-t +ε. As a corollary, there exists, for every integerq ≥ 3 and every constant ε 0, aq-query PCP for NP with amortized query complexity 1+ $ 1 + \sqrt{2/q} + \varepsilon $ + ε. Thisimproves upon the result of Samorodnitsky and Trevisan with respectto query efficiency, i.e., the relation between soundness and thenumber of queries. Although the improvement may seem moderatetheconstruction of Samorodnitsky and Trevisan has amortized querycomplexity 1 + 2-$ 1 + 2/\sqrt{q} + \varepsilon $we also show inthis article that combining our outer verifier with any naturalcandidate for a corresponding inner verifier gives a PCP that isless query efficient than the one we obtain.© 2008 WileyPeriodicals, Inc. Random Struct. Alg., 2008