Generalized Spanning Trees and Extensions

In this dissertation, network design problems where a global network interconnects at minimum cost local networks together (telecommunication, transportation networks) are considered. The generalized minimum spanning tree problem is defined on a graph where the vertex set is partitioned into clusters and a nonnegative cost is associated to each edge. It consists of finding a tree with minimum cost that includes exactly one vertex from each cluster. Several linear formulations are compared regarding their linear relaxation in order to determine a convenient formulation used in an exact algorithm. Several families of valid inequalities for this problem are shown to be facet-defining for the associated polytope. We have developed a Branchand-Cut algorithm that uses separation methods for these new classes of valid inequalities. A tabu heuristic is implemented providing an initial upper bound to the exact method. This exact method is tested on several types of instances: real world instances and randomly generated instances. Two extensions of the generalized minimum spanning tree problem are considered. The variant where the tree of minimum cost is allowed to contain at least one vertex in each cluster is studied. The problem of the maximum cost subgraph containing at most one vertex from each cluster is investigated.