Mathematical Programming

, Volume 47, Issue 1–3, pp 367–387 | Cite as

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 eachWi induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = ∪i=1k1{uv|u, v ∈ Wi,u ≠ v}. Given weightswe∈ℝ for alle ∈ E, the clique partitioning problem is to find a clique partitioningA ofG such that ∑e∈Awe 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 graphKn, i.e.,
is the convex hull of the incidence vectors of the clique partitionings ofKn. We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofKn 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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Copyright information

© The Mathematical Programming Society, Inc. 1990

Authors and Affiliations

  • M. Grötschel
    • 1
  • Y. Wakabayashi
    • 2
  1. 1.Institut für MathematikUniversität AugsburgAugsburgFR Germany
  2. 2.Instituto de Matemática e EsratísticaUniversidade de Sāo PauloSão PauloBrazil

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