The (minimum) rank of typical fooling set matrices

TL;DR

This paper investigates the typical minimum rank of fooling-set matrices with random zero-nonzero patterns, establishing bounds that depend on pattern density and field size, and highlighting open cases for further research.

Contribution

It provides probabilistic bounds on the minimum rank of random fooling-set matrices under various conditions, extending understanding of their typical behavior.

Findings

For certain densities, the minimum rank is linearly proportional to n.

The bounds depend on the pattern density p and the size of the field.

Open problems remain for cases with slowly vanishing p and large or infinite fields.

Abstract

A fooling-set matrix has nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the rank of such a matrix is at least $\sqrt n$. It is known that the bound is tight (up to a multiplicative constant). We ask for the "typical" minimum rank of a fooling-set matrix: For a fooling-set zero-nonzero pattern chosen at random, is the minimum rank of a matrix with that zero-nonzero pattern over a field $\mathbb F$ closer to its lower bound $\sqrt{n}$ or to its upper bound $n$? We study random patterns with a given density $p$, and prove an $\Omega(n)$ bound for the cases when: (a) $p$ tends to $0$ quickly enough, (b) $p$ tends to $0$ slowly, and $|\mathbb F|=O(1)$, (c) $p\in(0,1]$ is a constant. We have to leave open the case when $p\to 0$ slowly and $\mathbb F$ is a large or infinite field (e.g., $\mathbb F=GF(2^n)$, $F=\mathbb{R}$).