On Block Sensitivity and Fractional Block Sensitivity

TL;DR

This paper explores the relationship between block sensitivity and fractional block sensitivity of Boolean functions, improving the known bounds and providing a new family of functions with a tighter separation.

Contribution

It improves the constant factor in the known separation between fractional block sensitivity and block sensitivity, and introduces a new family of functions demonstrating this bound.

Findings

Enhanced the constant factor from previous bounds.

Constructed a new family of functions with tighter separation.

Established a more precise relationship between the measures.

Abstract

We investigate the relation between the block sensitivity $\text{bs}(f)$ and fractional block sensitivity $\text{fbs}(f)$ complexity measures of Boolean functions. While it is known that $\text{fbs}(f) = O(\text{bs}(f)^2)$, the best known separation achieves $\text{fbs}(f) = \left(\frac{1}{3\sqrt2} +o(1)\right) \text{bs(f)}^{3/2}$. We improve the constant factor and show a family of functions that give $\text{fbs}(f) = \left(\frac{1}{\sqrt6}-o(1)\right) \text{bs}(f)^{3/2}.$