A Note on a Communication Game

TL;DR

This paper discusses a communication game linked to the Sensitivity Conjecture, proposing new conjectures that could resolve longstanding questions in Boolean function complexity and query complexity.

Contribution

It introduces a new conjecture about a communication game that implies the Sensitivity Conjecture and related complexity hypotheses, including a stronger hypothesis and a query complexity question.

Findings

The main conjecture would imply the Sensitivity Conjecture.

A related conjecture could resolve a query complexity question about the Weak Parity problem.

Independent work by others also explored this game and its implications.

Abstract

We describe a communication game, and a conjecture about this game, whose proof would imply the well-known Sensitivity Conjecture asserting a polynomial relation between sensitivity and block sensitivity for Boolean functions. The author defined this game and observed the connection in Dec. 2013 - Jan. 2014. The game and connection were independently discovered by Gilmer, Kouck\'y, and Saks, who also established further results about the game (not proved by us) and published their results in ITCS '15 [GKS15]. This note records our independent work, including some observations that did not appear in [GKS15]. Namely, the main conjecture about this communication game would imply not only the Sensitivity Conjecture, but also a stronger hypothesis raised by Chung, F\"uredi, Graham, and Seymour [CFGS88]; and, another related conjecture we pose about a "query-bounded" variant of our communication game would suffice to answer a question of Aaronson, Ambainis, Balodis, and Bavarian [AABB14] about the query complexity of the "Weak Parity" problem---a question whose resolution was previously shown by [AABB14] to follow from a proof of the Chung et al. hypothesis.