The Euclidean $k$-Matching Problem is NP-hard

TL;DR

This paper proves that the Euclidean $k$-matching problem, which involves partitioning points into $k$-sets to minimize the sum of minimum spanning tree weights, is NP-hard for all fixed $k \\ge 3$, justifying heuristic use.

Contribution

It establishes the NP-hardness of the Euclidean $k$-matching problem for all fixed $k \\ge 3$, resolving an open problem and supporting heuristic approaches.

Findings

Proves NP-hardness for all fixed $k \\ge 3$ in Euclidean space.

Shows NP-hardness persists when trees are paths.

Provides theoretical justification for heuristic methods.

Abstract

Let $G$ be a complete edge-weighted graph on $n$ vertices. To each subset of vertices of $G$ assign the cost of the minimum spanning tree of the subset as its weight. Suppose that $n$ is a multiple of some fixed positive integer $k$. The $k$-matching problem is the problem of finding a partition of the vertices of $G$ into $k$-sets, that minimizes the sum of the weights of the $k$-sets. The case $k=3$ has been shown to be NP-hard [Johnsson et al.,1998]. In the Euclidean version, the vertices of $G$ are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean $k$-matching problem. We show that, for every fixed $k \ge 3$, the Euclidean $k$-matching is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case $k=3$. We also show that the problem remains NP-hard if the trees are required to be paths.