Computing Generalized Minimum Spanning Trees with Variable Neighborhood Search

In the generalized version of the classical Minimum Spanning Tree problem, the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. This problem plays, for example, a role in the design of backbones in larger communication networks. We present a Variable Neighborhood Search (VNS) approach for this problem which is based on two different neighborhood types working in complementary ways to maximize the efficiency gained from the VNS concept. Both types of neighborhoods are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. Tests on Euclidean and random instances indicate in particular on instances with many nodes per cluster significant advantages of our VNS over previously published metaheuristic approaches.