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An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem

Abstract

The parameterized node multiway cut problem is for a given graph to find a separator of size bounded by k whose removal separates a collection of terminal sets in the graph. In this paper, we develop an O(k4k n 3) time algorithm for this problem, significantly improving the previous algorithm of time \(O(4^{k^{3}}n^{5})\) for the problem. Our result gives the first polynomial time algorithm for the minimum node multiway cut problem when the separator size is bounded by O(log n).

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References

  1. 1.

    Boykov, Y., Veksler, O., Zabih, R.: Markov random fields with efficient approximations. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition, pp. 648–655, 1998

  2. 2.

    Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for multiway cut. J. Comput. Syst. Sci. 60, 564–574 (2000)

  3. 3.

    Chartrand, G., Lesniak, L.: Graphs & Digraphs, 2nd edn. Wadsworth/Cole Mathematics Series, Belmont (1986)

  4. 4.

    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. Manuscript (2007)

  5. 5.

    Cong, J., Labio, W., Shivakumar, N.: Multi-way VLSI circuit partitioning based on dual net representation. In: Proc. IEEE International Conference on Computer-Aided Design, pp. 56–62, 1994

  6. 6.

    Cunningham, W.: The optimal multiterminal cut problem. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 5, 105–120 (1991)

  7. 7.

    Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)

  8. 8.

    Downey, R., Fellows, M.: Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput. 24, 873–921 (1995)

  9. 9.

    Downey, R., Fellows, M.: Parameterized Complexity. Monograph in Computer Science. Springer, New York (1999)

  10. 10.

    Ford, L. Jr., Fulkerson, D.: Flows in Networks. Princeton University Press, Princeton (1962)

  11. 11.

    Karger, D., Levine, M.: Finding maximum flows in undirected graphs seems easier than bipartite matching. In: Proc. 30th Annual ACM Symposium on Theory of Computing, pp. 69–78, 1998

  12. 12.

    Karger, D., Klein, P., Stein, C., Thorup, M., Young, N.: Rounding algorithms for a geometric embedding of minimum multiway cut. In: Proc. 31th Annual ACM Symposium on Theory of Computing, pp. 668–678, 1999

  13. 13.

    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351, 394–406 (2006)

  14. 14.

    Naor, J., Zosin, L.: A 2-approximation algorithm for the directed multiway cut problem. SIAM J. Comput. 31, 477–482 (2001)

  15. 15.

    Stone, H.: Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans. Softw. Eng. 3, 85–93 (1977)

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Author information

Correspondence to Songjian Lu.

Additional information

A preliminary version of this paper was presented at The 10th Workshop on Algorithms and Data Structures (WADS 2007).

This work was supported in part by the National Science Foundation under the Grants CCR-0311590 and CCF-0430683.

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Chen, J., Liu, Y. & Lu, S. An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem. Algorithmica 55, 1–13 (2009). https://doi.org/10.1007/s00453-007-9130-6

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Keywords

  • Multiway cut problem
  • Parameterized algorithm
  • Fixed-parameter tractability
  • Minimum cut
  • Network flow