Volume 41, Number 3, July-September 2007Journées Polyèdres et Optimisation Combinatoire
|Page(s)||275 - 287|
|Published online||21 August 2007|
Clique partitioning of interval graphs with submodular costs on the cliques
Dep. of Operations Research, EGRES, Eötvös Lorand University, Pázmány Peter Setany. 1/C, 1117 Budapest, Hungary; email@example.com
2 CNRS, laboratoire Leibniz-IMAG, 46 avenue Félix Viallet, 38031 Grenoble Cedex, France; firstname.lastname@example.org
Accepted: 21 December 2006
Given a graph G = (V,E) and a “cost function” (provided by an oracle), the problem [PCliqW] consists in finding a partition into cliques of V(G) of minimum cost. Here, the cost of a partition is the sum of the costs of the cliques in the partition. We provide a polynomial time dynamic program for the case where G is an interval graph and f belongs to a subclass of submodular set functions, which we call “value-polymatroidal”. This provides a common solution for various generalizations of the coloring problem in co-interval graphs such as max-coloring, “Greene-Kleitman's dual”, probabilist coloring and chromatic entropy. In the last two cases, this is the first polytime algorithm for co-interval graphs. In contrast, NP-hardness of related problems is discussed. We also describe an ILP formulation for [PCliqW] which gives a common polyhedral framework to express min-max relations such as for perfect graphs and the polymatroid intersection theorem. This approach allows to provide a min-max formula for [PCliqW] if G is the line-graph of a bipartite graph and f is submodular. However, this approach fails to provide a min-max relation for [PCliqW] if G is an interval graphs and f is value-polymatroidal.
Mathematics Subject Classification: 90C27 / 05C15
Key words: Partition into cliques / Interval graphs / Circular arc graphs / Max-coloring / Probabilist coloring / Chromatic entropy / Partial q-coloring / Batch-scheduling / Submodular functions / Bipartite matchings / Split graphs
© EDP Sciences, ROADEF, SMAI, 2007
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