An Upper Bound on the GKS Game via Max Bipartite Matching

TL;DR

This paper improves the upper bound on the GKS communication game, which relates to the sensitivity conjecture, by developing a new technique to evaluate codeword strategies, reducing the known cost from approximately n^{0.4732} to n^{0.4696}.

Contribution

The paper introduces a novel method for assessing codeword strategies in the GKS game, leading to a tighter upper bound on the game's cost.

Findings

Improved the upper bound from O(n^{0.4732}) to O(n^{0.4696})

Developed a technique to identify viable codeword strategies in the GKS game

Enhanced understanding of the GKS game's complexity and strategy space

Abstract

The sensitivity conjecture is a longstanding conjecture concerning the relationship between the degree and sensitivity of a Boolean function. In 2015, a communication game was formulated by Justin Gilmer, Michal Kouck\'{y}, and Michael Saks to attempt to make progress on this conjecture. Andrew Drucker independently formulated this game. Shortly after the creation of the GKS game, Nisan Szegedy obtained a protocol for the game with a cost of $O(n^{.4732})$. We improve Szegedy's result to a cost of $O(n^{.4696})$ by providing a technique to identify whether a set of codewords can be used as a viable strategy in this game.