Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices

TL;DR

This paper explores the structural properties of the binary rank of Kronecker powers of matrices, introduces a column generation approach for fractional biclique partitioning, and reveals that fractional binary rank is not multiplicative, providing bounds and computational insights.

Contribution

It develops a novel column generation algorithm for fractional biclique partitioning and provides new bounds on the fractional binary rank of Kronecker powers, addressing open questions in matrix theory.

Findings

Fractional binary rank is not multiplicative under Kronecker product.

The asymptotic fractional binary rank of the Domino graph lies between 2 and 2.373.

The developed algorithm effectively manages exponential growth in biclique enumeration.

Abstract

We investigate structural properties of the binary rank of Kronecker powers of binary matrices, equivalently, the biclique partition numbers of the corresponding bipartite graphs. To this end, we engineer a Column Generation approach to solve linear optimization problems for the fractional biclique partition number of bipartite graphs, specifically examining the Domino graph and its Kronecker powers. We address the challenges posed by the double exponential growth of the number of bicliques in increasing Kronecker powers. We discuss various strategies to generate suitable initial sets of bicliques, including an inductive method for increasing Kronecker powers. We show how to manage the number of active bicliques to improve running time and to stay within memory limits. Our computational results reveal that the fractional binary rank is not multiplicative with respect to the Kronecker product. Hence, there are binary matrices, and bipartite graphs, respectively, such as the Domino, where the asymptotic fractional binary rank is strictly smaller than the fractional binary rank. While we used our algorithm to reduce the upper bound, we formally prove that the fractional biclique cover number is a lower bound, which is at least as good as the widely used isolating (or fooling set) bound. For the Domino, we obtain that the asymptotic fractional binary rank lies in the interval $[2,2.373]$. Since our computational resources are not sufficient to further reduce the upper bound, we encourage further exploration using more substantial computing resources or further mathematical engineering techniques to narrow the gap and advance our understanding of biclique partitions, particularly, to settle the open question whether binary rank and biclique partition number are multiplicative with respect to the Kronecker product.