Finding Spanning Trees with Perfect Matchings

TL;DR

This paper explores the computational complexity of finding spanning trees with perfect matchings, providing algorithms for special cases and proving NP-hardness for general and certain restricted graph classes.

Contribution

It introduces a greedy algorithm for specific weighted complete graphs and establishes NP-hardness results for broader graph classes, including bipartite and nonbipartite graphs.

Findings

Greedy algorithm works for complete graphs with two-value weights.

NP-hardness proven for complete, bipartite, and cubic planar graphs with limited weights.

Strongly balanced spanning trees are NP-hard to detect in subcubic planar graphs.

Abstract

We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight spanning tree among those containing a perfect matching. On the positive side, we design a simple greedy algorithm for the case when the graph is complete (or complete bipartite) and the edge weights take at most two values. On the negative side, the problem is NP-hard even when the graph is complete (or complete bipartite) and the edge weights take at most three values, or when the graph is cubic, planar, and bipartite and the edge weights take at most two values. We also consider an interesting variant. We call a tree strongly balanced if on one side of the bipartition of the vertex set with respect to the tree, all but one of the vertices have degree $2$ and the remaining one is a leaf. This property is a sufficient condition for a tree to have a perfect matching, which enjoys an additional property. When the underlying graph is bipartite, strongly balanced spanning trees can be written as matroid intersection, and this fact was recently utilized to design an approximation algorithm for some kind of connectivity augmentation problem. The natural question is its tractability in nonbipartite graphs. As a negative answer, it turns out NP-hard to test whether a given graph has a strongly balanced spanning tree or not even when the graph is subcubic and planar.