Upper and lower bounding strategies for the generalized minimum spanning tree problem

We address the generalized minimum spanning tree problem which requires spanning at least one vertex out of every set of disjoint vertices in a graph. We show that the geometric version of this problem is NP-hard, and we propose two stochastic heuristics. The first one is a very fast randomized greedy search algorithm and the second one is a genetic algorithm. Also, we investigate some existing integer programming formulations and present a new one. A new Lagrangian based lower bound is proposed and implemented to assess the performance of the heuristics. Computational experiments performed on a large set of randomly generated instances with up to 1000 vertices and 10,000 edges provide evidence of the good performance of the proposed heuristics.