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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.

Keywords

Planar Graph Steiner Tree Tree Decomposition Steiner Tree Problem Secondary Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

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