The hypocoloring problem: complexity and approximability results when the chromatic number is small

  • Authors:

  • Dominique de Werra;Marc Demange;Jerome Monnot;Vangelis Th. Paschos

  • Affiliations:

  • Ecole Polytechnique Fédérale de Lausanne, Switzerland;Dept. SID, ESSEC, France;LAMSADE, CNRS UMR 7024, Université Paris Dauphine, Paris, France;LAMSADE, CNRS UMR 7024, Université Paris Dauphine, Paris, France

  • Venue:
  • WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2004

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Abstract

We consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G=(V,E;w) where w(v)≥ 0, the goal consists in finding a partition ${\cal S}=(S_1,\ldots,S_k)$ of the node set of G into hypostable sets and minimizing ∑$_{i=1}^{k}$w(Si) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max {∑v∈Kw(v)| K∈S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Δ.