Models and branch-and-cut algorithms for the Steiner tree problem with revenues, budget and hop constraints

The Steiner tree problem with revenues, budget and hop constraints is a variant of the Steiner tree problem where, besides the costs associated with arcs, there are also revenues associated with the vertices. Budget and hop constraints impose limits on the total cost of the network and on the number of edges between any vertex and the root, respectively. This article introduces and compares several mathematical models for this problem and describes two branch-and-cut algorithms which solve to optimality instances with up to 500 vertices and 625 edges.