The classical graph edge coloring problem deals in coloring the edges of a given graph with minimum number of colors such that no two adjacent edges in the graph, get the same color in the proposed coloring. In the following work, we look at the other end of the spectrum where in our goal is to maximize the number of colors used for coloring the edges of the graph under some vertex specific constraints. We deal with the MAXIMUM EDGE COLORING problem which is defined as the following –For an integer q ≥2 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. The question is very well motivated by the problem of channel assignment in wireless networks. This problem is NP-hard for q ≥ 2, and has been well-studied from the point of view of approximation. This problem has not been studied in the parameterized context before. Hence as a next step, this thesis investigates the parameterized complexity of this problem where the standard parameter is the solution size. The main focus of the work is the special case of q=2 ,i.e. MAXIMUM EDGE 2-COLORING which is theoretically intricate and practically relevant in the wireless networks setting. We first show an exponential kernel for the MAXIMUM EDGE q-COLORING problem where q is a fixed constant and q ≥ 2.We do a more specific analysis for the kernel of the MAXIMUM EDGE 2-COLORING problem. The kernel obtained here is still exponential in size but is better than the kernel obtained for MAXIMUM EDGE q-COLORING problem in case of q=2. We then show a fixed parameter tractable algorithm for the MAXIMUM EDGE 2-COLORING problem with a running time of O*∗(kO(k)).We also show a fixed parameter tractable algorithm for the MAXIMUM EDGE q-COLORING problem with a running time of O∗(kO(qk) qO(k)). The fixed parameter tractability of the dual parametrization of the MAXIMUM EDGE 2-COLORING problem is established by arguing a linear vertex kernel for the problem. We also show that the MAXIMUM EDGE 2-COLORING problem remains hard on graphs where the maximum degree is a constant and also on graphs without cycles of length four. In both these cases, we obtain quadratic kernels. A closely related variant of the problem is the question of MAX EDGE{1,2-}COLORING. For this problem, the vertices in the input graph may have different qε,{1.2} values and the goal is to use at least k colors for the edge coloring of the graph such that every vertex sees at most q colors, where q is either one or two. We show that the MAX EDGE{1,2}-COLORING problem is W[1]-hard on graphs that have no cycles of length four.

Les problèmes d’ordonnancement de rendez-vous en tête-à-tête sont des problèmes dans lesquels des personnes souhaitent se rencontrer par deux lors de courts rendez-vous qui se déroulent lors d’une session unique. Dans cette thèse, nous référençons plusieurs applications de ce type de problèmes et proposons des notations qui généralisent les notations standards de problèmes d’ordonnancement α|β|γ. Nous nous intéressons en particulier à un cas dans lequel deux populations distinctes se rencontrent, des participants peuvent arriver en retard et des rencontres sont interdites. L’objectif est de minimiser le nombre maximal d’attentes des participants. Nous étudions dans un premier temps la complexité de ces problèmes : nous démontrons que plusieurs cas sans rencontre interdite sont polynomiaux et que le cas général est NP-complet au sens fort. Nous proposons ensuite des bornes inférieures. Puis nous développons plusieurs méthodes de résolution. Des modèles de programmation linéaire en nombres entiers et un modèle de programmation par contraintes sont tout d’abord proposés. Des règles de dominance permettant de limiter les symétries sont intégrées à ces modèles dans le but de limiter l’espace des solutions. Enfin, nous proposons une recherche à divergence limitée (limited discrepancy search) qui est une méthode approchée basée sur l’exploration d’un arbre de recherche tronqué. Dans cette méthode, nous exploitons le plus possible les propriétés de symétrie du problème pour faciliter la convergence vers une bonne solution. Toutes ces méthodes sont testées et comparées sur un ensemble de 300 instances générées aléatoirement d’après des paramètres réalistes. / One-to-one meeting scheduling problems are problems where a population of actors want to meet each other during short time slots that take place in a single session. In this thesis, we reference several applications of this type of problems found in the literature and introduce a notation extending the well-known scheduling notation α|β|γ. We are particularly interested in a case in which two distinct populations meet, participants may arrive late and some meetings are forbidden. The objective is to minimize the maximum number of participants waiting slots. First, we study the complexity of these problems: we show that several cases with no forbidden meeting are polynomial and that the general case is NP-complete in the strong sense. We then propose lower bounds. After that, we develop several resolution methods. Integer linear programming models and a constraint programming model are developed. To limit the solution space, we add dominance rules based on symmetries to these methods. Finally, we present a limited discrepancy search (i.e. an approximate method based on the exploration of a truncated tree search). In this method, we use as much as possible the symmetry properties of the problem to facilitate the convergence to a good solution. All these methods are tested and compared on a set of 300 randomly generated instances from realistic parameters.