On the average sensitivity and density of k-CNF formulas

Abstract

We study the relationship between the average sensitivity and density of k-CNF formulas via the isoperimetric function φ : [0, 1] → ℝ, φ(μ = max{AS(F)/CNF-width(F): E[F(x)] = μ}, where the maximum is taken over all Boolean functions F:{0, 1}* → {0, 1} over a finite number of variables and AS(F) is the average sensitivity of F. Building on the work of Boppana [1] and Traxler [2], and answering an open problem of O'Donnell, Amano [3] recently proved that φ(μ) ≤ 1 for all μ ∈ [0,1]. In this paper we determine φ exactly, giving matching upper and lower bounds. The heart of our upper bound is the Paturi-Pudlák-Zane (PPZ) algorithm for k-SAT [4], which we use in a unified proof that sharpens the three incomparable bounds of Boppana, Traxler, and Amano. We extend our techniques to determine φ when the maximum is taken over monotone Boolean functions F, further demonstrating the utility of the PPZ algorithm in isoperimetric problems of this nature. As an application we show that this yields the largest known separation between the average and maximum sensitivity of monotone Boolean functions, making progress on a conjecture of Servedio. Finally, we give an elementary proof that AS(F) ≤ log(s)(1 + o(1)) for functions F computed by an s-clause CNF, which is tight up to lower order terms. This sharpens and simplifies Boppana's bound of O(logs) obtained using Håstad's switching lemma. © 2013 Springer-Verlag.