Partitioning into common independent sets via relaxing strongly base orderability

TL;DR

This paper explores new relaxations of strongly base orderability in matroids to better understand the complexity of covering ground sets with common independent sets, offering algorithmic insights and extending existing theories.

Contribution

It introduces relaxations of strongly base orderability and new concepts for covering circuits, expanding the class of matroids with tractable properties and connecting to longstanding open problems.

Findings

Relaxed basis-exchange conditions define a new class of matroids.

Covering circuits by 2-regular or path graphs offers new algorithmic approaches.

Results relate to and extend existing conjectures on matroid coverings.

Abstract

The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids form a class for which a basis-exchange condition that is much stronger than the standard axiom is met. As a result, several problems that are open for arbitrary matroids can be solved for this class. In particular, Davies and McDiarmid showed that if both matroids are strongly base orderable, then the covering number of their intersection coincides with the maximum of their covering numbers. Motivated by their result, we propose relaxations of strongly base orderability in two directions. First we weaken the basis-exchange condition, which leads to the definition of a new, complete class of matroids with distinguished algorithmic properties. Second, we introduce the notion of covering the circuits of a matroid by a graph, and consider the cases when the graph is ought to be 2-regular or a path. We give an extensive list of results explaining how the proposed relaxations compare to existing conjectures and theorems on coverings by common independent sets.