Solving the prize-collecting steiner tree problem to optimality

The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears in the design of utility networks (eg. fiber optics or district heating) where profit generating customers and the network connecting them have to be chosen in the most profitable way. The primary focus of this paper is the construction of a branch-and-cut algorithm based on a directed graph model and connectivity inequalities corresponding to cuts in the graph. This enables us to efficiently separate sets of violated inequalities using a maximum flow algorithm. Our method solves all benchmark instances from the literature to optimality, including eight for which the optimum was not known. We also present optimal results on very large real world instances that represent fiber optic networks in a German city.