Le problème du séparateur de poids minimum : Complexité, Polyèdres et Algorithmes
Youcef Magnouche · HAL (Le Centre pour la Communication Scientifique Directe) · 2017
Given a graph G = (V U T, E) with V U T the set of vertices, where T is a set of terminals, and a weight function w, associated with the nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V U 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. We describe some valid inequalities and characterize when these inequalities define facets. Using these results, we develop a Branch-and-Cut algorithm for the problem. 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. Moreover, we propose three extended formulations for the problem and derive Branch-and-Price and Branch-and-Cut-and-Price algorithms. For each formulation we present a column generation scheme, the way to compute the dual bound, and the branching scheme. Finally, we discuss four variants of the multi-terminal vertex separator problem. We show that all these variants are NP-hard and for each one we give an integer programming formulation and present some class of valid inequalities.