Measuring Depth of Matroids

TL;DR

This paper introduces and systematically studies eight recursive depth measures for matroids and matrices, relating them to existing concepts like matroid branch-depth and tree-width, and compares them to classical graph depth measures.

Contribution

It unifies various recently introduced and scattered depth concepts into a general framework, proving fundamental properties and relationships among these measures.

Findings

Six of the eight measures are mutually functionally inequivalent.

Two measures are equivalent to matroid branch-depth and matroid tree-depth.

The measures coincide on matroids and matrices over any field.

Abstract

Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and a newly introduced natural depth counterpart of matroid tree-width. In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. Importantly, we also prove that these depth measures coincide on matroids and on matrices over any field, which is (somehow surprisingly) not a trivial task. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.