The multi-terminal vertex separator problem : Complexity, Polyhedra and Algorithms. (Le problème du séparateur de poids minimum : Complexité, Polyèdres et Algorithmes)

This thesis deals with the multi-terminal vertex separator problem. Given a graph G = (V ∪T,E) with V ∪T the set of vertices, where T is a set of terminals, and a weight function w : V → Z, associated with nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V ∪T into k+1 subsets {S, V1, . . . , Vk} such that there is no edge between two different subsets Vi and Vj, each Vi contains exactly one terminal and the weight of S is minimum. In this thesis, we consider the problem from a polyhedral point of view. We give two integer programming formulations for the problem, for one of them, we investigate the related polyhedron and discuss its polyhedral structure. We describe some valid inequalities and characterize when these inequalities define facets. We also derive separation algorithms for these inequalities. Using these results, we develop a Branch-and-Cut algorithm for the problem, along with an extensive computational study is presented. We also study the multi-terminal vertex separator polytope in the graphs decomposable by one node cutsets. If G is a graph that decomposes into G1 and G2, we show that the multi-terminal vertex separator polytope in G can be described from two linear systems related to G1 and G2. This gives rise to a technique for characterizing the multi-terminal vertex separator polytope in the graphs that are recursively decomposable. We also obtain a procedure to describe facets for this polytope and show that the multi-terminal vertex separator problem can also be decomposed. Applications of this technique are also discussed. Moreover, we propose three extended formulations for the problem and derive Branchand-Price and Branch-and-Cut-and-Price algorithms. For each formulation we present a column generation scheme to solve the linear relaxation, the way to compute the dual bound and the branching scheme. We present computational results and discuss the performance of each algorithm. Further, we discuss four variants of the multi-terminal vertex separator problem. We