Rainbow bases in matroids

TL;DR

This paper investigates the computational complexity of rainbow basis problems in matroids and related structures, proving certain cases are hard and providing bounds for covering matroids with rainbow bases.

Contribution

It extends hardness results to graphic matroids and specific digraph problems, and establishes bounds for covering matroids with rainbow bases under certain conditions.

Findings

Factorizing graphic matroids into rainbow bases is NP-hard.

Deciding if a digraph can be decomposed into bounded indegree spanning trees is NP-hard.

Matroids with a k-basis factorization can be covered by a bounded number of rainbow bases when partition classes are small.

Abstract

Recently, it was proved by B\'erczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of B\'erczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function $f$ such that every matroid that can be factorized into $k$ bases for some $k \geq 3$ can be covered by $f(k)$ rainbow bases if every partition class contains at most 2 elements.