A communication game related to the sensitivity conjecture

TL;DR

This paper introduces a novel two-player communication game related to the sensitivity conjecture in boolean functions, proposing new bounds and protocols to approach this longstanding open problem.

Contribution

It formulates a new communication game to attack the sensitivity conjecture and analyzes variants, providing protocols and bounds that advance understanding of the problem.

Findings

A lower bound of $n^{ ext{Omega}(1)}$ would imply the conjecture.

Protocols show the lower bound does not hold for two variants.

An improved upper bound of $ ext{sqrt}(n)$ on the game cost.

Abstract

One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that interpolates the function) is bounded above by some fixed power of its sensitivity (which is the maximum vertex degree of the graph defined on the inputs where two inputs are adjacent if they differ in exactly one coordinate and their function values are different). We propose an attack on the sensitivity conjecture in terms of a novel two-player communication game. A lower bound of the form $n^{\Omega(1)}$ on the cost of this game would imply the sensitivity conjecture. To investigate the problem of bounding the cost of the game, three natural (stronger) variants of the question are considered. For two of these variants, protocols are presented that show that the hoped for lower bound does not hold. These protocols satisfy a certain monotonicity property, and (in contrast to the situation for the two variants) we show that the cost of any monotone protocol satisfies a strong lower bound. There is an easy upper bound of $\sqrt{n}$ on the cost of the game. We also improve slightly on this upper bound.