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Given a complete Graph $G(V,E)$ with $|V|=kn$ and weights $w:E→N$ that satisfies "Triangular Inequality". That is, for any $v_1, v_2, v_3 \in V$,$$w(v_1,v_2)\le w(v_1,v_3)+w(v_3,v_2).$$ Can I find the minimum spanning forest $T_1,T_2,…,T_n$ where each tree $T_i$ has exactly $k$ vertices?

For $k=2$, the problem becomes the minimum-weight perfect matching problem, and we can use the Blossom algorithm. I wonder for $k>2$ this problem is P or NP, I am particularly interested in the case with $k=3$.

diego.0412
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  • I wonder whether a variant of the Kruskal minimal weight spanning tree algorithm works; order the edges by weight, then go through the edges starting at lowest weight, including an edge if it doesn't create a tree with more than three vertices. – Gerry Myerson Sep 10 '22 at 01:04

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