Mathematical Programming

, Volume 47, Issue 1–3, pp 367–387

# Facets of the clique partitioning polytope

• M. Grötschel
• Y. Wakabayashi
Article

## Abstract

A subsetA of the edge set of a graphG = (V, E) is called a clique partitioning ofG is there is a partition of the node setV into disjoint setsW1,⋯,Wk such that eachW i induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = ∪ i=1 k 1{uv|u, v ∈ W i ,u ≠ v}. Given weightsw e ∈ℝ for alle ∈ E, the clique partitioning problem is to find a clique partitioningA ofG such that ∑ e∈A w e is as small as possible. This problem—known to be-hard, see Wakabayashi (1986)—comes up, for instance, in data analysis, and here, the underlying graphG is typically a complete graph. In this paper we study the clique partitioning polytope of the complete graphK n , i.e., is the convex hull of the incidence vectors of the clique partitionings ofK n . We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofK n induce facets of. The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in Grötschel and Wakabayashi (1989).

## Key words

Polyhedral combinatorics clique partitioning data analysis

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