Improved Steiner Tree Algorithms for Bounded Treewidth

  • Markus Chimani
  • Petra Mutzel
  • Bernd Zey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

Abstract

We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in \(\ensuremath{\mathcal{O}(B_{\ensuremath{\textit{tw}}+2}^2 \cdot \ensuremath{\textit{tw}}\ \cdot |V|)}\) time, where \(\ensuremath{\textit{tw}}\) is the graph’s treewidth and the Bell number B k is the number of partitions of a k-element set. This is a linear time algorithm for graphs with fixed treewidth and a polynomial algorithm for \(\ensuremath{\textit{tw}} = \ensuremath{\mathcal{O}(\log|V|/\log\log|V|)}\).

While being faster than the previously known algorithms, our thereby used coloring scheme can be extended to give new, improved algorithms for the prize-collecting Steiner tree as well as the k-cardinality tree problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bateni, M., Chekuri, C., Ene, A., Hajiaghayi, M., Korula, N., Marx, D.: Prize-collecting Steiner problems on planar graphs. In: SODA, pp. 1028–1049. SIAM (2011)Google Scholar
  3. 3.
    Bateni, M., Hajiaghayi, M., Marx, D.: Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. In: STOC, pp. 211–220. ACM (2010)Google Scholar
  4. 4.
    Bateni, M., Hajiaghayi, M., Marx, D.: Prize-collecting network design on planar graphs. CoRR, abs/1006.4339 (2010)Google Scholar
  5. 5.
    Berend, D., Tassa, T.: Improved bounds on Bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics 30, 185–205 (2010)MathSciNetMATHGoogle Scholar
  6. 6.
    Bern, M.W., Lawler, E.L., Wong, A.L.: Linear-time computation of optimal subgraphs of decomposable graphs. J. Algorithms 8, 216–235 (1987)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Betzler, N.: Steiner tree problems in the analysis of biological networks. Master’s thesis, Universität Tübingen (2006)Google Scholar
  8. 8.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: Fast subset convolution. In: STOC, pp. 67–74. ACM (2007)Google Scholar
  9. 9.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11(1-2), 1–22 (1993)MathSciNetMATHGoogle Scholar
  10. 10.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bodlaender, H.L.: Treewidth: Structure and Algorithms. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 11–25. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Borradaile, G., Kenyon-Mathieu, C., Klein, P.: A polynomial-time approximation scheme for Steiner tree in planar graphs. In: SODA, pp. 1285–1294. SIAM (2007)Google Scholar
  13. 13.
    Borradaile, G., Klein, P., Mathieu, C.: An O(n logn) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms 5, 1–31 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chekuri, C., Ene, A., Korula, N.: Prize-collecting Steiner tree and forest in planar graphs. CoRR, abs/1006.4357 (2010)Google Scholar
  15. 15.
    Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. CoRR, abs/1103.0534 (2011)Google Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  17. 17.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1, 195–207 (1972)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM Journal on Applied Mathematics 32(4), 826–834 (1977)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gassner, E.: The Steiner forest problem revisited. J. Discrete Algorithms 8(2), 154–163 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Korach, E., Solel, N.: Linear time algorithm for minimum weight Steiner tree in graphs with bounded treewidth. Technical Report 632, Israel Institute of Technology (1990)Google Scholar
  21. 21.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  22. 22.
    Polzin, T., Daneshmand, S.: Practical Partitioning-Based Methods for the Steiner Problem. In: Àlvarez, C., Serna, M. (eds.) WEA 2006. LNCS, vol. 4007, pp. 241–252. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Prömel, H.J., Steger, A.: The Steiner Tree Problem. A Tour Through Graphs, Algorithms and Complexity. Vieweg Verlag (2002)Google Scholar
  24. 24.
    Ravi, R., Sundaram, R., Marathe, M.V., Rosenkrantz, D.J., Ravi, S.S.: Spanning trees short or small. In: SODA, pp. 546–555. SIAM (1994)Google Scholar
  25. 25.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Chimani
    • 1
  • Petra Mutzel
    • 2
  • Bernd Zey
    • 2
  1. 1.Institute of Computer ScienceFriedrich-Schiller-University of JenaGermany
  2. 2.Department of Computer ScienceTUDortmundGermany

Personalised recommendations