A Tighter Relation between Sensitivity and Certificate Complexity

TL;DR

This paper improves the upper bounds relating sensitivity and certificate complexity in decision tree complexity, advancing understanding of the sensitivity conjecture by analyzing the structure of sensitivity graphs.

Contribution

It provides a tighter upper bound on certificate complexity in terms of sensitivity, using structural analysis of sensitivity graphs, and extends results to functions with specific sensitivity properties.

Findings

New upper bound: C(f) ≤ (8/9 + o(1)) s(f) 2^{s(f)-1}

Deeper understanding of the structure of sensitivity graphs

Tighter relationship between C_0(f) and s_0(f) for functions with s_1(f)=2

Abstract

The sensitivity conjecture which claims that the sensitivity complexity is polynomially related to block sensitivity complexity, is one of the most important and challenging problem in decision tree complexity theory. Despite of a lot of efforts, the best known upper bound of block sensitivity, as well as the certificate complexity, are still exponential in terms of sensitivity: $bs(f)\leq C(f)\leq\max\{2^{s(f)-1}(s(f)-\frac{1}{3}),s(f)\}$. In this paper, we give a better upper bound of $bs(f)\leq C(f)\leq(\frac{8}{9} + o(1))s(f)2^{s(f) - 1}$. The proof is based on a deep investigation on the structure of the sensitivity graph. We also provide a tighter relationship between $C_0(f)$ and $s_0(f)$ for functions with $s_1(f)=2$.