Semi-Definite positive Programming Relaxations for Graph K-Coloring in Frequency Assignment

Issue		RAIRO Oper. Res. Volume 35, Number 2, April-June 2001


Page(s)		211 - 228
DOI		10.1051/ro:2001112

DOI: 10.1051/ro:2001112

RAIRO Oper. Res. 35 (2001) 211-228

Semi-Definite positive Programming Relaxations for Graph K $_{\bf n}$ -Coloring in Frequency Assignment

Philippe Meurdesoif¹ and Benoît Rottembourg²

¹ Projet NUMOPT, INRIA Rhône-Alpes, 655 avenue de l'Europe, 38330 Montbonnot-Saint-Martin, France; (Philippe.Meurdesoif@inrialpes.fr)
² Direction des Technologies Nouvelles, Bouygues, 1 avenue Eugène Freyssinet, 78061 Saint-Quentin-en-Yvelines, France; (brottemb@challenger.bouygues.fr)

(Received May, 1999.)

Abstract
In this paper we will describe a new class of coloring problems, arising from military frequency assignment, where we want to minimize the number of distinct n-uples of colors used to color a given set of n-complete-subgraphs of a graph. We will propose two relaxations based on Semi-Definite Programming models for graph and hypergraph coloring, to approximate those (generally) NP-hard problems, as well as a generalization of the works of Karger et al. for hypergraph coloring, to find good feasible solutions with a probabilistic approach.

Résumé
Dans cet article, nous décrivons une nouvelle classe de problèmes de coloration rencontrés en Allocation de Fréquences militaire : nous voulons minimiser le nombre de n-uplets distincts utilisés pour colorier un ensemble doné de n-cliques d'un graphe. Pour approcher ces problèmes généralement NP-difficiles, nous proposons deux relaxations basées sur les modélisations semi-définies de la coloration de graphes et d'hypergraphes, ainsi qu'une généralisation des travaux de Karger et al. à la coloration d'hypergraphes, pour trouver de bonnes solutions faisables par une approche probabiliste.

Key words: Discrete optimization, semidefinite programming frequency assignment, graph coloring, hypergraph coloring.

© EDP Sciences 2001

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