The prize-collecting generalized minimum spanning tree problem

The prize-collecting generalized minimum spanning tree (PC-GMST) is defined as follows: Given an undirected graph G = (V, E), with node set V, edge set E, cost vector c ∈ R|E| + on the edges E, prize vector p ∈ R|V | + on the nodes V, and a set of K mutually exclusive and exhaustive node sets V1, ..., VK (i.e., V1 ⋂ Vj = ∅, if i 6= j, and Kk=1 Vk = V ), find a minimum cost tree spanning exactly one node from each cluster. One application of the PC-GMST problem is in the regional connection of local area networks (LAN), where several LANs in a region need to be connected with each other. For this purpose, one gateway node needs to be identified within each LAN, and the gateway nodes are to be connected via minimum spanning tree. Additionally, nodes within the same cluster are competing to be selected as gateway nodes, and each node offers a certain compensation (prize) if selected. This problem represents a generalization of the GMST problem first introduced in [3]. In this paper, we explore several exact and heuristic procedures for the PC-GMST problem, and discuss computational experiments.