Improper colouring of weighted grid and hexagonal graphs

Jean-Claude Bermond (1) , Frédéric Havet (1) , Florian Huc (2) , Claudia Linhares Sales (3)

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
2 Centre Universitaire d'Informatique
3 LIA - Laboratórios de PesquIsa em ComputAção

We study a weighted improper colouring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colours (frequencies), such that the set of vertices having a given colour induces a graph of degree at most k (the case k = 0 corresponds to a proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such colouring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same.

Domaines: Mathématique discrète [cs.DM]; Algorithme et structure de données [cs.DS]

Jean-Claude Bermond, Frédéric Havet, Florian Huc, Claudia Linhares Sales. Improper colouring of weighted grid and hexagonal graphs. Discrete Mathematics, Algorithms and Applications, 2010, 2 (3), pp.395-411. ⟨10.1142/S1793830910000747⟩. ⟨inria-00526530⟩