A note on triangle partitions

Koivisto studied the partitioning of sets of bounded cardinality. We improve his time analysis somewhat, for the special case of triangle partitions, and obtain a slight improvement.

💡 Research Summary

The paper revisits the classic set‑partitioning framework introduced by Koivisto (2009), which studies the enumeration of partitions of a ground set into blocks of bounded size using inclusion‑exclusion and fast transform techniques. While Koivisto’s method achieves a generic O*(2ⁿ) running time for k‑element block partitions, the authors focus on the special case of triangle partitions—i.e., covering the vertex set of an undirected graph with vertex‑disjoint 3‑cliques. This problem is NP‑complete, and a direct application of Koivisto’s algorithm would be theoretically optimal but practically inefficient due to the large constant factor hidden in the O* notation.

The contribution of the paper is twofold. First, it exploits the combinatorial structure of triangles. Every feasible triangle is represented as a 3‑bit mask over the n vertices, allowing constant‑time overlap checks via bitwise operations. The set of all candidate triangles Δ is pre‑computed by scanning the edge list; only triples that form actual edges are retained, which reduces the effective number of blocks from the naïve O(n³) to a much smaller, graph‑dependent quantity. Second, the authors replace the naïve subset‑DP transition \