Mathematical Programming

, Volume 105, Issue 2, pp 427–449

An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem


    • Vienna University of Technology
  • René Weiskircher
    • Vienna University of Technology
  • Ulrich Pferschy
    • University of Graz
  • Gunnar W. Klau
    • Vienna University of Technology
  • Petra Mutzel
    • Vienna University of Technology
  • Matteo Fischetti
    • DEIUniversity of Padova

DOI: 10.1007/s10107-005-0660-x

Cite this article as:
Ljubić, I., Weiskircher, R., Pferschy, U. et al. Math. Program. (2006) 105: 427. doi:10.1007/s10107-005-0660-x


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 frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way.

Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems.

We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values.


Branch-and-CutSteiner ArborescencePrize CollectingNetwork Design
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© Springer-Verlag Berlin Heidelberg 2005