The Influence of Preprocessing on Steiner Tree Approximations

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9486)

Abstract

Given an edge-weighted graph G and a node subset R, the Steiner tree problem asks for an R-spanning tree of minimum weight. There are several strong approximation algorithms for this NP-hard problem, but research on their practicality is still in its early stages.

In this study, we investigate how the behavior of approximation algorithms changes when applying preprocessing routines first. In particular, the shrunken instances allow us to consider algorithm parameterizations that have been impractical before, shedding new light on the algorithms’ respective drawbacks and benefits.

This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We thank Mihai Popa for implementations of reduction tests, and Google for funding him through the Google Summer of Code 2014 program. We also thank the authors of [8] for making their exact solver available, and Renato Werneck for detailed logs of their experiments in [17].

Supplementary material

Open image in new window

References

  1. 1.
    Beyer, S., Chimani, M.: Steiner tree 1.39-approximation in practice. In: Hliněný, P., Dvořák, Z., Jaroš, J., Kofroň, J., Kořenek, J., Matula, P., Pala, K. (eds.) MEMICS 2014. LNCS, vol. 8934, pp. 60–72. Springer, Heidelberg (2014) Google Scholar
  2. 2.
    Beyer, S., Chimani, M.: Strong Steiner Tree Approximations inPractice. Submitted to Journal (2014). http://arxiv.org/abs/1409.8318
  3. 3.
    Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: Steiner tree approximation via iterative randomized rounding. J. ACM 60(1), 6:1–6:33 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chimani, M., Woste, M.: Contraction-based Steiner tree approximations in practice. In: Asano, T., Nakano, S., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 40–49. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Ciebiera, K., Godlewski, P., Sankowski, P., Wygocki, P.: Approximation Algorithms for Steiner Tree Problems Based on Universal Solution Frameworks. arXiv abs/1410.7534 (2014). http://arxiv.org/abs/1410.7534
  6. 6.
    11th DIMACS Challenge. http://dimacs11.cs.princeton.edu. Bounds 12 September 2014
  7. 7.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1(3), 195–207 (1971)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fischetti, M., Leitner, M., Ljubic, I., Luipersbeck, M., Monaci, M., Resch, M., Salvagnin, D., Sinnl, M.: Thinning out Steiner trees: a node-based model for uniform edge costs. In: 11th DIMACS Challenge (2014)Google Scholar
  9. 9.
    Goemans, M.X., Olver, N., Rothvoß, T., Zenklusen, R.: Matroids and integrality gaps for hypergraphic steiner tree relaxations. In: STOC 2012, pp. 1161–1176. ACM (2012)Google Scholar
  10. 10.
    Junger, M., Thienel, S.: The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Softw.: Pract. Exp. 30, 1325–1352 (2000)MATHGoogle Scholar
  11. 11.
    Koch, T., Martin, A., Voß, S.: SteinLib: An Updated Library on Steiner Tree Problems in Graphs. ZIB-Report 00–37 (2000). http://steinlib.zib.de
  12. 12.
    Kou, L.T., Markowsky, G., Berman, L.: A fast algorithm for Steiner trees. Acta Informatica 15, 141–145 (1981)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ljubic, I., Weiskircher, R., Pferschy, U., Klau, G.W., Mutzel, P., Fischetti, M.: An algorithmic framework for the exact solution of the prize-collecting Steiner tree problem. Math. Program. 105(2–3), 427–449 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mehlhorn, K.: A faster approximation algorithm for the steiner problem in graphs. Inf. Process. Lett. 27(3), 125–128 (1988)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. In: STOC 1988, 229–234. ACM (1988)Google Scholar
  16. 16.
    Poggi de Aragão, M., Ribeiro, C.C., Uchoa, E., Werneck, R.F.: Hybrid local search for the steiner problem in graphs. In: MIC 2001 (2001)Google Scholar
  17. 17.
    Pajor, T., Uchoa, E., Werneck, R.F.: A robust and scalable algorithm for the Steiner problem in graphs. In: 11th DIMACS Challenge (2014)Google Scholar
  18. 18.
    Polzin, T., Vahdati Daneshmand, S.: Improved algorithms for the Steiner problem in networks. Discrete Appl. Math. 112(1–3), 263–300 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Polzin, T., Vahdati Daneshmand, S.: The Steiner tree challenge: an updated study. In: 11th DIMACS Challenge (2014)Google Scholar
  20. 20.
    Robins, G., Zelikovsky, A.: Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math. 19(1), 122–134 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Takahashi, H., Matsuyama, A.: An approximate solution for the Steiner problem in graphs. Math. Jpn. 24, 573–577 (1980)MathSciNetMATHGoogle Scholar
  22. 22.
    Zelikovsky, A.: An 11/6-approximation algorithm for the Steiner problem on graphs. Ann. Discrete Math. 51, 351–354 (1992)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zelikovsky, A.: A faster approximation algorithm for the Steiner tree problem in graphs. Inf. Process. Lett. 46(2), 79–83 (1993)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zelikovsky, A.: An 11/6-approximation algorithm for the network Steiner problem. Algorithmica 9(5), 463–470 (1993)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zelikovsky, A.: Better Approximation Bounds for the Network and Euclidean Steiner Tree Problems. Technical report. CS-96-06, University of Virginia (1995)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of OsnabrückOsnabrückGermany

Personalised recommendations