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...
| Authors: | , , , , , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 2026 |
| Country: | España |
| Institution: | Universidad de Sevilla (US) |
| Repository: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:dnet:idus________::10f6e93336a3674abc563b48fca41497 |
| Online Access: | https://hdl.handle.net/11441/186792 https://doi.org/10.1016/j.comgeo.2026.102267 |
| Access Level: | Open access |
| Keyword: | Euclidean 3-matching NP-completeness Planar 1-in-3 SAT |
| Summary: | 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 (sets of k elements), that minimizes the sum of the weights of the k-sets. The case of 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 ≥ 3, the Euclidean k-matching problem 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 of k =3. We also show that the problem remains NP-hard if the trees are required to be paths. |
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