Approches Métaheuristiques à base de Population pour la Coloration de Graphes

Abstract

Vertex coloring is one of the most studied problems in graph theory. It consists of coloring vertices of a graph by a minimum number of colors called the chromatic number such that no two adjacent vertices have the same color. Finding the chromatic number is proved to be NPcomplete for general graphs. According to the problem nature and the covered applications, many other constraints can be added to the main definition of the graph coloring. The consideration of such constraints leads to new graph coloring kinds and varieties, such as: total coloring, list coloring, multi coloring and T-coloring. In the first part of this thesis, we tackle the strict strong graph coloring. This coloring includes a dominance relation between color classes and graph vertices. This problem has been solved for trees and proved to be NP-complete for general graphs. To solve this NP hard problem, a good approach is to use evolutionary computation that allows finding optimal values on complex and large search spaces. In this thesis, we introduce two basic approaches for solving the problem for general graphs. The first approach searches an optimal SSColoring using legal coloring space. Here, we use a particular encoding, a specific initialization and crossover operators that preserve the problem properties. Contrary to the first approach, the second one consists in building first non-valid SSColoring configurations ensuring the dominance without ensuring the proper coloring property. Then, a correction operator is used, to improve the quality of the selected parents and the produced offspring until obtaining a valid SSColoring. Afterwards these approaches are then optimized including a new constructive optimization operator that helps heuristically to converge toward the global optimal solution. In the second part of this thesis, we investigate the use of a recent nature inspired algorithm, called the flower pollination algorithm (FPA), to solve the classical graph coloring problem. More precisely, we introduce two contributions: Since the original version of the FPA algorithm was developed for continuous valued spaces, in the first approach, we present a discrete basic version of the FPA algorithm to solve the problem. Contrary to the first approach that omits invalid solutions, the second approach adapts the efficient constructive method, called RLF (Recursive Largest First), to deal with invalid solutions that may occur during the FPA process. The approach is then improved by including swapping and inversion strategies. Through experiments, we show that the proposed approaches give better results compared to the existing algorithms and, in some cases, may reach the exact solutions.

La coloration de graphes est un problème classique de la théorie des graphes et de l‟optimisation combinatoire. Il s'agit de colorer les sommets d'un graphe afin que deux sommets adjacents n‟aient pas la même couleur. Ce problème a donné lieu à de nombreuses variantes telles que la coloration forte et la coloration forte stricte où une relation de dominance entre les sommets du graphe est présente. Généralement, la coloration forte stricte est une coloration propre telle que chaque sommet du graphe domine au moins une classe de couleurs non vide. Le problème de la coloration forte stricte étant NP-difficile, l‟utilisation des méthodes exactes pour le résoudre s‟avère inappropriée. Les méthodes approchées, ou plus exactement les algorithmes évolutionnaires, représentent une alternative qui donne de bonnes solutions en un temps de résolution raisonnable. Dans notre travail, nous proposons d‟abord deux nouveaux algorithmes évolutionnaires qui donnent, après leurs exécutions, une coloration forte stricte d‟un graphe général avec un nombre de couleurs minimal.