Mathematical Programming

, Volume 105, Issue 2–3, pp 427–449 | Cite as

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

  • Ivana Ljubić
  • René Weiskircher
  • Ulrich Pferschy
  • Gunnar W. Klau
  • Petra Mutzel
  • Matteo Fischetti
Article

Abstract

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.

Keywords

Branch-and-Cut Steiner Arborescence Prize Collecting Network Design 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aneja, Y.P.: An integer linear programming approach to the Steiner problem in graphs. Networks 10, 167–178 (1980)MATHMathSciNetGoogle Scholar
  2. 2.
    Bachhiesl, P., Prossegger, M., Paulus, G., Werner, J., Stögner, H.: Simulation and optimization of the implementation costs for the last mile of fiber optic networks. Networks and Spatial Economics 3 (4), 467–482 (2003)Google Scholar
  3. 3.
    Beasley, J.E.: An SST-based algorithm for the Steiner problem in graphs. Networks 19, 1–16 (1989)MATHMathSciNetGoogle Scholar
  4. 4.
    Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.: A note on the prize-collecting traveling salesman problem. Mathematical Programming 59, 413–420 (1993)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Canuto, S.A., Resende, M.G.C., Ribeiro, C.C.: Local search with perturbations for the prize-collecting Steiner tree problem in graphs. Networks 38, 50–58 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cherkassky, B.V., Goldberg, A.V.: On implementing push-relabel method for the maximum flow problem. Algorithmica 19, 390–410 (1997)MATHMathSciNetGoogle Scholar
  7. 7.
    Chopra, S., Gorres, E., Rao, M.R.: Solving a Steiner tree problem on a graph using a branch and cut. ORSA Journal on Computing, 4, 320–335 (1992)MATHGoogle Scholar
  8. 8.
    Chopra, S., Rao, M.R.: The Steiner tree problem I: Formulations, compositions and extension of facets. Mathematical Programming 64, 209–229 (1994)MATHMathSciNetGoogle Scholar
  9. 9.
    Dongarra, J.J.: Performance of various computers using standard linear equations software (linpack benchmark report). Technical Report CS-89-85, University of Tennessee, 2004Google Scholar
  10. 10.
    Duin, C.W., Volgenant, A.: Some generalizations of the Steiner problem in graphs. Networks 17 (2), 353–364 (1987)MathSciNetGoogle Scholar
  11. 11.
    Engevall, S., Göthe-Lundgren, M., Värbrand, P.: A strong lower bound for the node weighted Steiner tree problem. Networks 31 (1), 11–17 (1998)MathSciNetGoogle Scholar
  12. 12.
    Feofiloff, P., Fernandes, C.G., Ferreira, C.E., Pina, J.C.: Primal-dual approximation algorithms for the prize-collecting Steiner tree problem. 2003 (submitted)Google Scholar
  13. 13.
    Fischetti, M.: Facets of two Steiner arborescence polyhedra. Mathematical Programming 51, 401–419 (1991)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Goemans, M.X.: The Steiner tree polytope and related polyhedra. Mathematical Programming 63, 157–182 (1994)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Goemans, M.X., Williamson, D.P.: The primal-dual method for approximation algorithms and its application to network design problems. In: Hochbaum, D.S. (ed) Approximation algorithms for NP-hard problems, P. W. S. Publishing Co., 1996, pp 144–191Google Scholar
  16. 16.
    Gutin, G., Punnen, A. (eds) The traveling salesman problem and its variations. Kluwer, 2002Google Scholar
  17. 17.
    Hackner, J.: Energiewirtschaftlich optimale Ausbauplanung kommunaler Fernwärmesysteme. PhD thesis, Vienna University of Technology, Austria, 2004Google Scholar
  18. 18.
    Johnson, D.S., Minkoff, M., Phillips, S.: The prize-collecting Steiner tree problem: Theory and practice. In: Proceedings of 11th ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, 2000, pp 760–769Google Scholar
  19. 19.
    Klau, G.W., Ljubić, I., Moser, A., Mutzel, P., Neuner, P., Pferschy, U., Weiskircher, R.: Combining a memetic algorithm with integer programming to solve the prize-collecting Steiner tree problem. In: Deb, K. (ed), Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2004), volume 3102 of LNCS, Springer-Verlag, 2004, pp 1304–1315Google Scholar
  20. 20.
    Koch, T., Martin, A.: Solving Steiner tree problems in graphs to optimality. Networks, 32, 207–232 (1998)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Ljubić, I., Weiskircher, R., Pferschy, U., Klau, G.W., Mutzel, P., Fischetti, M.: Solving the prize-collecting Steiner tree problem to optimality. In: Proceedings of the Seventh Workshop on Algorithm Engineering and Experiments (ALENEX 05). SIAM, 2005 (to appear)Google Scholar
  22. 22.
    Lucena, A., Resende, M.G.C.: Strong lower bounds for the prize-collecting Steiner problem in graphs. Discrete Applied Mathematics 141, 277–294 (2004)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Minkoff, M.: The prize-collecting Steiner tree problem. Master's thesis, MIT, May, 2000Google Scholar
  24. 24.
    Segev, A.: The node-weighted Steiner tree problem. Networks 17, 1–17 (1987)MATHMathSciNetGoogle Scholar
  25. 25.
    Uchoa, E.: Reduction tests for the prize-collecting Steiner problem. Technical Report RPEP Vol.4 no.18, Universidade Federal Fluminense, Engenharia de Produção, Niterói, Brazil, 2004Google Scholar
  26. 26.
    Wong, R.T.: A dual ascent based approach for the Steiner tree problem in directed graphs. Mathematical Programming 28, 271–287 (1984)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ivana Ljubić
    • 1
  • René Weiskircher
    • 1
  • Ulrich Pferschy
    • 2
  • Gunnar W. Klau
    • 1
  • Petra Mutzel
    • 1
  • Matteo Fischetti
    • 3
  1. 1.Vienna University of TechnologyViennaAustria
  2. 2.University of GrazGrazAustria
  3. 3.DEIUniversity of PadovaPadovaItaly

Personalised recommendations