Finding Spanning Trees with Perfect Matchings

TL;DR

This paper studies the fusion of two fundamental graph structures by examining the problem of finding a minimum-weight spanning tree that contains a perfect matching (PMST) and a strongly balanced spanning tree (SBST). It provides a precise tractability boundary: MinPMST is solvable in polynomial time for complete or complete bipartite graphs when edge weights take at most two values, but becomes NP-hard under modest relaxations (three weights, or certain sparse/dense configurations); SBST is NP-hard even for subcubic planar graphs. The authors present a simple greedy augmentation-based algorithm for the tractable two-weight complete cases and use reductions from Hamiltonian cycle and planar 3-SAT to establish hardness results, including a matroid-intersection viewpoint for SBST in bipartite graphs. Together, these results delineate the complexity landscape for combined spanning-tree and matching constraints and motivate future work on approximations and fixed-parameter methods.

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.

Figures (5)

• Figure 1: Construction of $\tilde{G}$ from $G$. The middle graph is the intermediate graph just after replacing each edge with two disjoint paths. Dashed lines represent edges of weight $0$, and solid lines represent edges of weight $1$. In this example, $\tilde{e}$ and $\tilde{e}'$ are derived from $e = f_{u, 1} = f_{v, 3}$.
• Figure 2: The variable gadget for a variable $x_i$. In any strongly balanced spanning tree $T$ (if exists), the black vertices will be in $V_T^+$ (with degree constraint $2$), the white vertices will be in $V_T^-$ (without degree constraint), and the checkered vertices can be in either side, alternately from $u_i$; the lower figure illustrates a subtree corresponding to an assignment with $x_i = 1$.
• Figure 3: The whole structure of the variable gadgets. In any strongly balanced spanning tree $T$ (if exists), the black vertices will be in $V_T^+$ (with degree constraint $2$ except for $s_1$, which will be the unique leaf in $V_T^+$) and the white vertices will be in $V_T^-$ (without degree constraint).
• Figure 4: The clause gadget for a clause $C_j = (y_{j,1} \vee y_{j,2} \vee y_{j,3})$. In any strongly balanced spanning tree $T$ (if exists), the black vertices will be in $V_T^+$ (with degree constraint $2$), the white vertices will be in $V_T^-$ (without degree constraint), and the checkered vertices can be in either side depending on the situation in the corresponding variable gadgets; the right figure illustrates an example of a subtree corresponding to the situation when $C_j$ is satisfied by $y_{j,2} = 1$.
• Figure 5: An alternative five-vertex gadget for each leaf. The lower vertex $\ell$ corresponds to an original leaf. In any strongly balanced spanning tree $T$ (if exists), the black vertices will be in $V_T^+$ (with degree constraint $2$), and the white vertices will be in $V_T^-$ (without degree constraint), except for exactly one of these gadgets (that corresponds to $s_1$ in Figure \ref{['fig:SBST_path']}).

Theorems & Definitions (12)

• Lemma 2.1: Norose and Yamaguchi norose2024approximation
• Theorem 2.2
• Theorem 2.3
• Lemma 3.2
• proof
• Remark 3.3
• Theorem 3.4: Akiyama, Nishizeki, and Saito akiyama1980np
• Claim 3.5
• proof
• Theorem 4.2: Lichtenstein lichtenstein1982planar