Base Modulus for Matroid Truncation, Strength, and Fractional Arboricity

TL;DR

This paper explores the universal density in matroid theory, its behavior under truncation, and its connections to concepts like strength and fractional arboricity, with applications to graph optimization problems.

Contribution

It characterizes the universal density for all truncations of a matroid, introduces a new KL divergence-based characterization, and extends the notion of strictly homogeneous matroids.

Findings

Universal density determined for every matroid truncation.

New KL divergence-based characterization of universal density.

Insights into strength, fractional arboricity, and graph optimization problems.

Abstract

In [27], we provided results on the $p$-modulus of the family of all bases of matroids and showed that it recovers various concepts in matroid theory, including strength, fractional arboricity, and principal partitions. In particular, the unique optimal density $\eta^*$ that arises for $p$-modulus, which we will refer to as universal density from now on, was shown to recover the concept of lexicographical base in polymatroids. Since truncation is a fundamental operation in matroid theory, it is natural to ask how the universal density behaves under matroid truncation. In this paper, we first provide the universal density of every truncation of a given matroid; equivalently, we determine the principal partition for every matroid truncation. Next, we give a new characterization of the universal density using the Kullback--Leibler divergence. Furthermore, we study the notion of strictly homogeneous matroids, generalizing the corresponding notion in graphs from [6]. We also offer several insights related to strength, fractional arboricity, and give the set of probability mass functions (pmfs) for bases that induce the universal density in a simple case. Finally, this paper also addresses two optimization problems for graph structures, particularly those involving edge-disjoint spanning trees and forest edge-coverings.