The generalized minimum spanning tree problem: Polyhedral analysis and branch-and-cut algorithm

This article presents a branch-and-cut algorithm for the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose vertex set is partitioned into clusters, the GMSTP consists of determining a minimum cost tree including exactly one vertex per cluster. Applications of the GMSTP are encountered in telecommunications. An integer linear programming formulation is presented and new classes of valid inequalities are developed, several of which are proved to be facet defining. A branch-and-cut algorithm and a tabu search heuristic are developed. Extensive computational experiments show that instances involving up to 160 or 200 vertices can be solved to optimality, depending on whether edge costs are Euclidean or random.