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Gowers uniformity, influence of variables, and PCPs

Alex Samorodnitsky,L. Trevisan

Published 2005 in Symposium on the Theory of Computing

ABSTRACT

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 2t-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 cq/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 Δ 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.

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