Some combinatorial optimization problems in graphs with applications in telecommunications and tomography

The common point between the different chapters of the present work is graph theory. We investigate some well known graph theory problems, and some which arise from more specific applications. In the first chapter, we deal with the maximum stable set problem, and provide some new graph classes, where it can be solved in polynomial time. Those classes are hereditary, i.e. characterized by a list of forbidden induced subgraphs. The algorithms proposed are purely combinatorial. The second chapter is devoted to the study of a problem linked to security purposes in mobile telecommunication networks. The particularity is that there is no central authority guaranteeing security, but it is actually managed by the users themselves. The network is modelled by an oriented graph, whose vertices represent the users, and whose arcs represent public key certificates. The problem is to associate to each vertex a subgraph with some requirements on the size of the subgraphs, the number of times a vertex is taken in a subgraph and the connectivity between any two users as they put their subgraphs together. Constructive heuristics are proposed, bounds on the optimal solution and a tabu search are described and tested. The third chapter is on the problem of reconstructing an image, given its projections in terms of the number of occurrences of each color in each row and each column. The case of two colors is known to be polynomially solvable, it is NP-complete with four or more colors, and the complexity status of the problem with three colors is open. An intermediate case between two and three colors is shown to be solvable in polynomial time. The last two chapters are about graph (vertex-)coloring. In the fourth, we prove a result which brings a large collection of NP-hard subcases, characterized by forbidden induced subgraphs. In the fifth chapter, we approach the problem with the use of linear programming. Links between different formulations are pointed out, and some families of facets are characterized. In the last section, we study a branch and bound algorithm, whose lower bounds are given by the optimal value of the linear relaxation of one of the exposed formulations. A preprocessing procedure is proposed and tested.