Communication is bounded by root of rank

TL;DR

This paper establishes a near-quadratic improvement in the deterministic communication complexity for total boolean functions based on the rank of their associated matrices, linking rank to chromatic number.

Contribution

It proves a new upper bound on communication complexity and chromatic number in terms of matrix rank, improving previous bounds significantly.

Findings

Deterministic communication complexity is O(√r · log r) for functions of rank r.

Chromatic number of graphs with adjacency matrix rank r is at most 2^{O(√r · log r)}.

Provides a nearly quadratic improvement over prior bounds.

Abstract

We prove that any total boolean function of rank $r$ can be computed by a deterministic communication protocol of complexity $O(\sqrt{r} \cdot \log(r))$. Equivalently, any graph whose adjacency matrix has rank $r$ has chromatic number at most $2^{O(\sqrt{r} \cdot \log(r))}$. This gives a nearly quadratic improvement in the dependence on the rank over previous results.