2022/03/24 by Haviv, Ishay, Parnas, Michal
#Combinatorics (math.CO) #Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2203.13073
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\widetildeΩ(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\widetildeΩ(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, Göös, 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\widetildeΩ(log k).