Facets of the clique partitioning polytope |
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Authors: | M Grötschel Y Wakabayashi |
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Institution: | (1) Institut für Mathematik, Universität Augsburg, 8900 Augsburg, FR Germany;(2) Instituto de Matemática e Esratística, Universidade de So Paulo, 01498 São Paulo, SP, Brazil |
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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 setsW
1,,W
k 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
for alle E, the clique partitioning problem is to find a clique partitioningA ofG such that
eA
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). |
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Keywords: | Polyhedral combinatorics clique partitioning data analysis |
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