Sensitivity versus Certificate Complexity of Boolean Functions

TL;DR

This paper improves bounds relating sensitivity, block sensitivity, and certificate complexity of Boolean functions, advancing understanding of the sensitivity conjecture by providing tighter inequalities and examining specific sensitivity cases.

Contribution

It presents a tighter upper bound on certificate complexity in terms of sensitivity, utilizing recent structural theorems, and explores these relations for functions with small 1-sensitivity.

Findings

Improved upper bound: $C(f) \,\leq\, \max(2^{s(f)-1}(s(f)-\frac{1}{3}), s(f))$

Established relations between complexity measures for functions with small 1-sensitivity

Enhanced understanding of the sensitivity conjecture through structural analysis

Abstract

Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been notoriously hard to obtain even exponential bounds. Since block sensitivity is known to be polynomially related to certificate complexity, an equivalent of proving this conjecture would be showing that certificate complexity is polynomially related to sensitivity. Previously, it has been shown that $bs(f) \leq C(f) \leq 2^{s(f)-1} s(f) - (s(f)-1)$. In this work, we give a better upper bound of $bs(f) \leq C(f) \leq \max\left(2^{s(f)-1}\left(s(f)-\frac 1 3\right), s(f)\right)$ using a recent theorem limiting the structure of function graphs. We also examine relations between these measures for functions with small 1-sensitivity $s_1(f)$ and arbitrary 0-sensitivity $s_0(f)$.