Gowers uniformity, influence of variables, and PCPs

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+e, for arbitrarily small e>0. For values of q of the form 2t-1, the soundness error is (q+1)/2q+e.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 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 uniformity and influence might have other applications.