On the Binary and Boolean Rank of Regular Matrices

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 Ω(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 Ω(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 ö ös, 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 Ω(log k) .