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

Abstract

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.

The prize-collecting generalized minimum spanning tree problem

“…When all the prizes are equal to zero (or equivalently are exactly equal to each other within a node set in the partition), this problem corresponds to the generalized minimum spanning tree problem, that has been studied recently by several groups of researchers (see Feremans et al 2004;Golden et al 2005;Pop 2004). Since the GMST problem is NP-hard, it implies (by restriction) that the PCGMST problem is also NP-hard.…”

Generalized network design polyhedra

“…We remark that the odd cycle inequalities for the GMTSP, presented by Feremans et al [8], are the special case of the odd ring inequalities obtained when D = C. Moreover, it can be shown that the 'cycle' inequalities for the partial constraint satisfaction polytope, presented in Koster et al [12], are equivalent to odd ring inequalities, in the sense that the cycle and odd ring inequalities define the same facets of that polytope. The odd clique inequalities, on the other hand, appear to be entirely new.…”

“…If |V k | > 1 and {v} is a cluster, then the SVUB is dominated by the cluster constraints x v = 1 and x(V k ) = 1, together with the VUBs y uv ≤ x u for all u ∈ V k . Now, Proposition 7 of [8] states that, in all other cases, the SVUB inequality defines a facet of the GMTSP polytope. The GMSTP polytope is contained in P(G), and has dimension |V | + |E| − m − 1, because it satisfies the equation y(E) = m − 1 in addition to the equations satisfied by P(G).…”

“…For example, Fischetti et al [9] focussed on GTSP polyhedra, whereas Feremans et al [8] focussed on GMSTP polyhedra. Our goal in this article is to show that it is possible, to a certain extent, to derive polyhedral results for all GNDPs simultaneously.…”

“…When d = c/2 , they reduce to the "odd clique matching" inequalities, introduced by Feremans et al [8] in the context of the GMSTP. On the other hand, when d = 1, they reduce to the odd cycle inequalities, also due to Feremans et al [8], that we mentioned in the previous subsection.…”