Weighted improper colouring

Julio Araujo 1, 2 Jean-Claude Bermond 1 Frédéric Giroire 1 Frédéric Havet 1 Dorian Mazauric 1, 3 Remigiusz Modrzejewski 1
1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 MAESTRO - Models for the performance analysis and the control of networks
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this paper, we study a colouring problem motivated by a practical frequency assignment problem and, up to our best knowledge, new. In wireless networks, a node interferes with other nodes, the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph $(G,w)$ where the weight function $w$ on the edges of $G$ represents the noise (interference) between the two end-vertices. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted $t$-improper $k$-colouring of $(G,w)$ is a $k$-colouring of the nodes of $G$ (assignment of $k$ frequencies) such that the interference at each node does not exceed the threshold $t$. We consider here the Weighted Improper Colouring problem which consists in determining the weighted $t$-improper chromatic number defined as the minimum integer $k$ such that $(G,w)$ admits a weighted $t$-improper $k$-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where, given a number $k$ of colours, we want to determine the minimum real $t$ such that $(G,w)$ admits a weighted $t$-improper $k$-colouring. We show that both problems are NP-hard and first present general upper bounds for both problems; in particular we show a generalisation of Lovász's Theorem for the weighted $t$-improper chromatic number. Motivated by the original application, we then study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes at distance two. We derive the weighted $t$-improper chromatic number for all values of $t$.
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Julio Araujo, Jean-Claude Bermond, Frédéric Giroire, Frédéric Havet, Dorian Mazauric, et al.. Weighted improper colouring. Journal of Discrete Algorithms, Elsevier, 2012, 16, pp.53-66. ⟨10.1016/j.jda.2012.07.001⟩. ⟨hal-00747755⟩

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