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

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.