A Study of the Binary and Boolean Rank of Matrices with Small Constant Real Rank

TL;DR

This paper investigates the relationship between real, binary, and Boolean ranks of small-rank matrices, providing tight bounds and characterizing extremal cases for ranks 1 to 4.

Contribution

It establishes tight bounds and characterizations for the binary and Boolean ranks of matrices with small real rank, revealing the maximal gaps and unique extremal matrices.

Findings

Tight bounds for Boolean and binary ranks when real rank is 1 to 4.

Identification of unique extremal matrices achieving maximal rank gaps.

Connection to minimal biclique covers in bipartite graphs.

Abstract

We initiate the study of the binary and Boolean rank of $0,1$ matrices that have a small rank over the reals. The relationship between these three rank functions is an important open question, and here we prove that when the real rank $d$ is a small constant, the gap between the real and the binary and Boolean rank is a small constant. We give tight upper and lower bounds on the Boolean and binary rank of matrices with real rank $1 \leq d \leq 4$, as well as determine the size of the largest isolation set in each case. Furthermore, we prove that for $d = 3,4$, the circulant matrix defined by a row with $d-1$ consecutive ones followed by $d-1$ zeros, is the only matrix of size $(2d-2)\times (2d-2)$ with real rank $d$ and Boolean and binary rank and isolation set of size $2d-2$, and this matrix achieves the maximal gap possible between the real and the binary and Boolean rank for these values of $d$. Our results can also be interpreted in other equivalent terms, such as finding the minimal number of bicliques needed to partition or cover the edges of a bipartite graph whose reduced adjacency matrix has real rank $1 \leq d \leq 4$. We use a combination of combinatorial and algebraic techniques combined with the assistance of a computer program.