TY - GEN

T1 - Gowers uniformity, influence of variables, and PCPs

AU - Samorodnitsky, Alex

AU - Trevisan, Luca

PY - 2006

Y1 - 2006

N2 - We return to the study of the relation of query complexity and soundness in probabilistically checkable proofs. We present a PCP verifier for languages that are Unique-Games-Hard and such that the verifier makes q queries, has almost perfect completeness, and has soundness error at most 2q/2q + ε, for arbitrarily small ε > 0. For values of q of the form 2 t - 1, the soundness error is (q + 1)/2q + ε. Charikar et al. show that there is a constant c such that for every language that has a verifier of query complexity q, and a ratio of soundness error to completeness smaller than cg/2q is decidable in polynomial time. Up to the value of the multiplicative constant and to the validity of the Unique Games Conjecture, our result is therefore tight. As a corollary, we show that approximating the Maximum Independent Set problem in graphs of degree A within a factor better than Δ/(log δ)c is Unique-Games-Hard for a certain constant c > 0. Our main technical results are (i) a connection between the Gowers uniformity of a Boolean function and the influence of its variables and (ii) the proof that "Gowers uniform" functions pass the "hypergraph linearity test" approximately with the same probability of a random function. The connection between Gowers uniformity and influence might have other applications.

AB - We return to the study of the relation of query complexity and soundness in probabilistically checkable proofs. We present a PCP verifier for languages that are Unique-Games-Hard and such that the verifier makes q queries, has almost perfect completeness, and has soundness error at most 2q/2q + ε, for arbitrarily small ε > 0. For values of q of the form 2 t - 1, the soundness error is (q + 1)/2q + ε. Charikar et al. show that there is a constant c such that for every language that has a verifier of query complexity q, and a ratio of soundness error to completeness smaller than cg/2q is decidable in polynomial time. Up to the value of the multiplicative constant and to the validity of the Unique Games Conjecture, our result is therefore tight. As a corollary, we show that approximating the Maximum Independent Set problem in graphs of degree A within a factor better than Δ/(log δ)c is Unique-Games-Hard for a certain constant c > 0. Our main technical results are (i) a connection between the Gowers uniformity of a Boolean function and the influence of its variables and (ii) the proof that "Gowers uniform" functions pass the "hypergraph linearity test" approximately with the same probability of a random function. The connection between Gowers uniformity and influence might have other applications.

KW - Influence of variables

KW - Linearity test

KW - Probabilistically checkable proofs

UR - http://www.scopus.com/inward/record.url?scp=33748120485&partnerID=8YFLogxK

U2 - 10.1145/1132516.1132519

DO - 10.1145/1132516.1132519

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AN - SCOPUS:33748120485

SN - 1595931341

SN - 9781595931344

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 11

EP - 20

BT - STOC'06

PB - Association for Computing Machinery

T2 - 38th Annual ACM Symposium on Theory of Computing, STOC'06

Y2 - 21 May 2006 through 23 May 2006

ER -