The prize-collecting generalized minimum spanning tree problem

We introduce the prize-collecting generalized minimum spanning tree problem. In this problem a network of node clusters needs to be connected via a tree architecture using exactly one node per cluster. Nodes in each cluster compete by offering a payment for selection. This problem is NP-hard, and we describe several heuristic strategies, including local search and a genetic algorithm. Further, we present a simple and computationally efficient branch-and-cut algorithm. Our computational study indicates that our branch-and-cut algorithm finds optimal solutions for networks with up to 200 nodes within two hours of CPU time, while the heuristic search procedures rapidly find near-optimal solutions for all of the test instances.