On the complexity of packing rainbow spanning trees

TL;DR

This paper investigates the computational complexity of packing rainbow spanning trees, showing NP-completeness in specific cases and extending classical results on disjoint arborescences.

Contribution

It identifies the NP-completeness of packing rainbow spanning trees in certain graph classes, extending Edmonds' classical results and addressing open questions in graph decomposition.

Findings

Deciding two disjoint rainbow spanning trees is NP-complete.

Complexity holds even when the graph is a union of two spanning trees with two edges per color.

Provides a negative answer to a question on decomposition of oriented k-partition-connected digraphs.

Abstract

One of the most important questions in matroid optimization is to find disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases. B\'erczi and Schwarcz showed that the problem is hard in general, therefore identifying the borderline between tractable and intractable instances is of interest. In the present paper, we study the special case when one of the matroids is a partition matroid while the other one is a graphic matroid. This setting is equivalent to the problem of packing rainbow spanning trees, an extension of the problem of packing arborescences in directed graphs which was answered by Edmonds' seminal result on disjoint arborescences. We complement his result by showing that it is NP-complete to decide whether an edge-colored graph contains two disjoint rainbow spanning trees. Our complexity result holds even for the very special case when the graph is the union of two spanning trees and each color class contains exactly two edges. As a corollary, we give a negative answer to a question on the decomposition of oriented $k$-partition-connected digraphs.