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Multicut Algorithms via Tree Decompositions

  • Reinhard Pichler
  • Stefan Rümmele
  • Stefan Woltran
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)

Abstract

Various forms of multicut problems are of great importance in the area of network design. In general, these problems are intractable. However, several parameters have been identified which lead to fixed-parameter tractability (FPT). Recently, Gottlob and Lee have proposed the treewidth of the structure representing the graph and the set of pairs of terminal vertices as one such parameter. In this work, we show how this theoretical FPT result can be turned into efficient algorithms for optimization, counting, and enumeration problems in this area.

Keywords

Child Node Tree Decomposition Terminal Vertex Discrete Apply Mathematic Terminal Pair 
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.
    Costa, M.C., Létocart, L., Roupin, F.: Minimal multicut and maximal integer multiflow: A survey. European Journal of Operational Research 162, 55–69 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Călinescu, G., Fernandes, C.G., Reed, B.A.: Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width. J. Alg. 48, 333–359 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bentz, C.: A simple algorithm for multicuts in planar graphs with outer terminals. Discrete Applied Mathematics 157, 1959–1964 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  7. 7.
    Bousquet, N., Daligault, J., Thomassé, S., Yeo, A.: A polynomial kernel for multicut in trees. In: Proc. STACS 2009. LIPIcs, vol. 3, pp. 183–194 (2009)Google Scholar
  8. 8.
    Marx, D., Razgon, I.: Constant ratio fixed-parameter approximation of the edge multicut problem. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 647–658. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351, 394–406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Xiao, M.: Simple and improved parameterized algorithms for multiterminal cuts. Theory Comput. Syst. (to appear, 2010)Google Scholar
  11. 11.
    Guo, J., Hüffner, F., Kenar, E., Niedermeier, R., Uhlmann, J.: Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs. European Journal of Operational Research 186, 542–553 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bentz, C.: On the complexity of the multicut problem in bounded tree-width graphs and digraphs. Discrete Applied Mathematics 156, 1908–1917 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gottlob, G., Lee, S.T.: A logical approach to multicut problems. Inf. Process. Lett. 103, 136–141 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: Handbook of Theor. Comp. Sci., vol. B, pp. 193–242. Elsevier Science Publishers, Amsterdam (1990)Google Scholar
  15. 15.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pichler, R., Rümmele, S., Woltran, S.: Multicut algorithms via tree decompositions. Technical Report DBAI-TR-2010-67, Technische Universität Wien (2010)Google Scholar
  17. 17.
    Kloks, T.: Treewidth: Computations and Approximations. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  18. 18.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    van den Eijkhof, F., Bodlaender, H.L., Koster, A.M.C.A.: Safe reduction rules for weighted treewidth. Algorithmica 47, 139–158 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51, 255–269 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Reinhard Pichler
    • 1
  • Stefan Rümmele
    • 1
  • Stefan Woltran
    • 1
  1. 1.Vienna University of TechnologyAustria

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