Results 1  10
of
13
Weighted Improper Colouring
 22TH INTERNATIONAL WORKSHOP, IWOCA 2011
, 2011
"... 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 the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topog ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
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 the 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 where the weights on the edges represent the noise (interference) between the two endnodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted timproper kcolouring of G is a kcolouring of the nodes of G (assignment of k frequencies) such that the interference at each node does not exceed some threshold t. The Weighted Improper Colouring problem, that we consider here consists in determining the weighted timproper chromatic number defined as the minimum integer k such that G admits a weighted timproper kcolouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where given a number k of colours (frequencies) we want to determine the minimum real t such that G admits a weighted timproper kcolouring. We show that both problems are NPhard and first present general upper bounds; in particular we show a generalisation of Lovász’s Theorem for the weighted timproper chromatic number. We then show how to transform an instance of the Threshold Improper Colouring problem into another equivalent one
Exact Algorithms For L(2, 1)Labeling of Graphs
, 2007
"... The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G = (V, E) into an interval of integers{ 0,..., k} is an L(2, 1)labeling of G of span k if any two adjacent vertices are mapped onto integers ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G = (V, E) into an interval of integers{ 0,..., k} is an L(2, 1)labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k> = 4, deciding the existence of such a labeling is an NPcomplete problem. We present exact exponential time algorithms that are faster than the naive O((k + 1)n) algorithm that would try all possible mappings. The improvement is best seen in the first NPcomplete case of k = 4 here the running time of our algorithm is O(1.3006n). Furthermore we show that dynamic programming can be used to establish an O(3.8730n) algorithm to compute an optimal L(2, 1)labeling.
DOI: 10.1016/j.jda.2012.07.001 Weighted Improper Colouring. ✩
, 2012
"... 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 topogra ..."
Abstract
 Add to MetaCart
(Show Context)
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 endvertices. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted timproper kcolouring of (G, w) is a kcolouring 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 timproper chromatic number defined as the minimum integer k such that (G, w) admits a weighted timproper kcolouring. 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 timproper kcolouring. We first present general upper bounds for both problems; in particular we show a generalisation of Lovász’s Theorem for the weighted timproper 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 timproper chromatic number for all values of t.
IPFP6015964 AEOLUS Algorithmic Principles for Building Efficient Overlay Computers Deliverable D1.0.2
"... 4 SP management activities 12 ..."
(Show Context)
Aide totale demandée
"... Titre du projet en français Titre du projet en anglais CSD principale CSD secondaire (si interdisciplinarité) Algorithmes de graphes paramétrés et exacts Parameterized and exact graph algorithms ..."
Abstract
 Add to MetaCart
Titre du projet en français Titre du projet en anglais CSD principale CSD secondaire (si interdisciplinarité) Algorithmes de graphes paramétrés et exacts Parameterized and exact graph algorithms
ProjectTeam Mascotte Méthodes Algorithmiques, Simulation et Combinatoire pour l’OpTimisation des
"... c t i v it y e p o r t 2009 Table of contents ..."
(Show Context)
Weighted Improper Colouring.I
, 2012
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract
 Add to MetaCart
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.