Advertisement

Algorithms for Multiterminal Cuts

  • Mingyu Xiao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)

Abstract

Given a graph G = (V,E) with n vertices and m edges, and a subset T of l vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of k edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for l ≥ 3 but well-known to be polynomial-time solvable for l = 2 by the flow technique. In this paper, we show that Edge Multiterminal Cut is polynomial-time solvable for k = O(logn) by presenting an O(2 k lT(n,m)) algorithm, where T(n,m) = O( min (n 2/3,m 1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. We also give two algorithms for Vertex Multiterminal Cut that run in O(l k T(n,m)) time and O((k!)2 T(n,m)) time respectively. The former one indicates that Vertex Multiterminal Cut is solvable in polynomial time for l being a constant and k = O(logn), and the latter one improves the best known algorithm of running time \(O(4^{k^3}n^{O(1)})\). When l = 3, we show that the running times can be improved to O(1.415 k T(n,m)) for Edge Multiterminal Cut and O(2.059 k T(n,m)) for Vertex Multiterminal Cut. Furthermore, we present a simple idea to solve another important problem Multicut by finding minimum multiterminal cuts. Our algorithms for Multicuts are also faster than the previously best algorithm.

Based on a notion farthest minimum isolating cut, we present some properties for Multiterminal Cuts, which help shed light on the structure of optimal cut problems, and enables us to design efficient algorithms for Multiterminal Cuts, as well as some other related cut problems.

Keywords

Graph Algorithm Multiterminal Cut Multicut Fixed Parameter Tractability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Calinescu, G., Fernandes, C., Reed, B.: Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width. J. Algorithms 48(2), 333–359 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Calinescu, G., Karloff, H.J., Rabani, Y.: An improved approximation algorithm for multiway cut. Journal of Computer and Systems Sciences 60(3), 564–574 (2000); a preliminary version appeared in STOC 1998 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, J., Liu, Y., Lu, S.: An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, Springer, Heidelberg (2007)Google Scholar
  4. 4.
    Costa, M.-C., Létocart, L., Roupin, F.: Minimal multicut and maximal integer multiflow: A survey. European Journal of Operational Research 162(1), 55–69 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23(4), 864–894 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dinic, E.: Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Math. Dokl. 11, 1277–1280 (1970)Google Scholar
  7. 7.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Feige, U., Mahdian, M.: Finding small balanced separators. In: Proceedings of the 38th Annual ACM symposium on Theory of Computing (STOC 2006) (2006)Google Scholar
  9. 9.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. J. Algorithms 50(1), 49–61 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM 45(5), 783–797 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldschmidt, O., Hochbaum, D.: A polynomial algorithm for the k-cut problem for fixed k. Mathematics of Operations Research 19(1), 24–37 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Guo, J., Hüffner, F., Kenar, E., Niedermeier, R., Uhlmann, J.: Complexity and exact algorithms for multicut. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Ford, J.R., Fullkerson, D.R.: Flows in networks. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  14. 14.
    Karger, D.R., Klein, P.N., Stein, C., Thorup, M., Young, N.E.: Rounding algorithms for a geometric embedding relaxation of minimum multiway cut. In: Proceedings of the 31th Annual ACM Symposium on Theory of Computing (STOC 1999) (1999)Google Scholar
  15. 15.
    Marx, D.: The closest substring problem with small distances. In: Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS 2005) (2005)Google Scholar
  16. 16.
    Marx, D.: Parameterized graph separation problems. Theoret. Comput. Sci. 351(3), 394–406 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Naor, J., Zosin, L.: A 2-approximation algorithm for the directed multiway cut problem. SIAM J. Comput. 31(2), 477–482 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Yeh, W.-C.: A simple algorithm for the planar multiway cut problem. J. Algorithms 39(1), 68–77 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mingyu Xiao
    • 1
  1. 1.Department of Computer Science and EngineeringThe Chinese University of Hong KongHong Kong SARChina

Personalised recommendations