On the Binary and Boolean Rank of Regular Matrices

TL;DR

This paper constructs regular 0-1 matrices with specific binary and Boolean ranks, resolving longstanding questions and providing counterexamples in graph theory and communication complexity.

Contribution

It demonstrates the existence of regular matrices with contrasting binary and Boolean ranks, settling open problems and conjectures in linear algebra and graph theory.

Findings

Existence of regular matrices with binary rank k and Boolean cover complexity k^{ ilde{ ilde{ ext{Omega}}}( ext{log} k)}.

Counterexamples to the Alon-Saks-Seymour conjecture using regular graphs.

Connections to communication complexity and combinatorial matrix theory.

Abstract

A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers $k$, there exists a square regular $0,1$ matrix with binary rank $k$, such that the Boolean rank of its complement is $k^{\widetilde{\Omega}(\log k)}$. Equivalently, the ones in the matrix can be partitioned into $k$ combinatorial rectangles, whereas the number of rectangles needed for any cover of its zeros is $k^{\widetilde{\Omega}(\log k)}$. This settles, in a strong form, a question of Pullman (Linear Algebra Appl., 1988) and a conjecture of Hefner, Henson, Lundgren, and Maybee (Congr. Numer., 1990). The result can be viewed as a regular analogue of a recent result of Balodis, Ben-David, G\"{o}\"{o}s, Jain, and Kothari (FOCS, 2021), motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers $k$, there exists a regular graph with biclique partition number $k$ and chromatic number $k^{\widetilde{\Omega}(\log k)}$.