An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem
- 655 Downloads
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.
KeywordsBranch-and-Cut Steiner Arborescence Prize Collecting Network Design
Unable to display preview. Download preview PDF.
- 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
- 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
- 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
- 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.Gutin, G., Punnen, A. (eds) The traveling salesman problem and its variations. Kluwer, 2002Google Scholar
- 17.Hackner, J.: Energiewirtschaftlich optimale Ausbauplanung kommunaler Fernwärmesysteme. PhD thesis, Vienna University of Technology, Austria, 2004Google Scholar
- 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.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
- 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
- 23.Minkoff, M.: The prize-collecting Steiner tree problem. Master's thesis, MIT, May, 2000Google Scholar
- 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