Tighter Relations Between Sensitivity and Other Complexity Measures

TL;DR

This paper improves upper bounds relating sensitivity to other Boolean function complexity measures, advancing understanding of the longstanding sensitivity conjecture.

Contribution

It introduces new bounds that tighten the relationship between sensitivity and measures like degree and certificate complexity, marking progress after a decade.

Findings

Deg(f)^{1-o(1)}=O(2^{s(f)})

C(f) < 2^{s(f)-1} s(f)

First improvements in sensitivity bounds in ten years

Abstract

Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is polynomially related to other major complexity measures. Despite much attention to the problem and major advances in analysis of Boolean functions in the past decade, the problem remains wide open with no positive result toward the conjecture since the work of Kenyon and Kutin from 2004. In this work, we present new upper bounds for various complexity measures in terms of sensitivity improving the bounds provided by Kenyon and Kutin. Specifically, we show that deg(f)^{1-o(1)}=O(2^{s(f)}) and C(f) < 2^{s(f)-1} s(f); these in turn imply various corollaries regarding the relation between sensitivity and other complexity measures, such as block sensitivity, via known results. The gap between sensitivity and other complexity measures remains exponential but these results are the first improvement for this difficult problem that has been achieved in a decade.