A GRASP-based approach to the generalized minimum spanning tree problem

Given a multipartite graph G the generalized minimum spanning tree problem is to find a tree of minimal cost that includes a vertex from each part. This paper proposes several versions of the GRASP metaheuristic for the problem. The GRASP approach is based on constructive heuristics as well as on additional improvement mechanisms such as path-relinking and iterated local search. Several computational experiments are performed over a set of existing instances. A cut generation algorithm is proposed that is able to find lower bounds, based on a formulation for Steiner’s problem in directed graphs. The computational results show that the best versions of the GRASP approach use improvement mechanisms. The solutions found are better than most of the known solutions in the literature and require significantly less computer time. Furthermore, a set of rules is defined for pre-processing the instances, based on the Bottleneck distance concept. Using those rules, it was possible to reduce the size of the instances to an average of 14% of the number of edges in relation to the original graphs.