On Structural Rank and Resilience of Sparsity Patterns

TL;DR

This paper studies the resilience of sparsity patterns in matrices, specifically how their structural rank is affected by adding or removing entries, using max-flow algorithms on bipartite graphs.

Contribution

It introduces a polynomial-time method to analyze and enhance the resilience of sparsity patterns' structural rank through graph-based algorithms.

Findings

Algorithms for resilience analysis are polynomial-time solvable.

The approach effectively quantifies how many entries can be added or removed.

Resilience measures can be computed efficiently for large patterns.

Abstract

A sparsity pattern in $\mathbb{R}^{n \times m}$, for $m\geq n$, is a vector subspace of matrices admitting a basis consisting of canonical basis vectors in $\mathbb{R}^{n \times m}$. We represent a sparsity pattern by a matrix with $0/\star$-entries, where $\star$-entries are arbitrary real numbers and $0$-entries are equal to $0$. We say that a sparsity pattern has full structural rank if the maximal rank of matrices contained in it is $n$. In this paper, we investigate the degree of resilience of patterns with full structural rank: We address questions such as how many $\star$-entries can be removed without decreasing the structural rank and, reciprocally, how many $\star$-entries one needs to add so as to increase the said degree of resilience to reach a target. Our approach goes by translating these questions into max-flow problems on appropriately defined bipartite graphs. Based on these translations, we provide algorithms that solve the problems in polynomial time.