Algorithmica

, Volume 62, Issue 3–4, pp 787–806 | Cite as

Divide-and-Conquer Algorithms for Partitioning Hypergraphs and Submodular Systems

  • Kazumasa Okumoto
  • Takuro Fukunaga
  • Hiroshi Nagamochi
Article

Abstract

The submodular system k-partition problem is a problem of partitioning a given finite set V into k non-empty subsets V1,V2,…,Vk so that \(\sum_{i=1}^{k}f(V_{i})\) is minimized where f is a non-negative submodular function on V. In this paper, we design an approximation algorithm for the problem with fixed k. We also analyze the approximation factor of our algorithm for the hypergraph k-cut problem, which is a problem contained by the submodular system k-partition problem.

Keywords

Divide-and-conquer algorithm Hypergraph Multicut Submodular function 

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References

  1. 1.
    Frank, A.: Applications of submodular functions. In: Surveys in Combinatorics. Cambridge London Mathematical Society Lecture Notes Series, vol. 187, pp. 85–136 (1993) CrossRefGoogle Scholar
  2. 2.
    Fujishige, S.: Submodular Function and Optimization. North-Holland, Amsterdam (1991) Google Scholar
  3. 3.
    Fukunaga, T.: Computing minimum multiway cuts in hypergraphs from hypertree packings. In: Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 6080, pp. 15–28. Springer, Berlin (2010) Google Scholar
  4. 4.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. J. Algorithms 50, 49–61 (2004) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Gasieniec, L., Jansson, J., Lingas, A., Óstlin, A.: On the complexity of constructing evolutionary trees. J. Comb. Optim. 3, 183–197 (1999) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. ACM 35, 921–940 (1988) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Goldschmidt, O., Hochbaum, D.: A polynomial algorithm for the k-cut problem for fixed k. Math. Oper. Res. 19, 24–37 (1994) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Iwata, S.: Submodular function minimization. Math. Program. 112, 45–64 (2008) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Kamidoi, Y., Yoshida, N., Nagamochi, H.: A deterministic algorithm for finding all minimum k-way cuts. SIAM J. Comput. 36, 1329–1341 (2006) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Karger, D.R., Stein, C.: A new approach to the minimum cut problem. J. ACM 43, 601–640 (1996) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Internal Report B 96-02, Bericht FU Berlin Fachbereich Mathematik und Informatik (1995) Google Scholar
  12. 12.
    Lawler, E.L.: Cutsets and partitions of hypergraphs. Networks 3, 275–285 (1973) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Mak, W.-K., Wong, D.F.: A fast hypergraph min-cut algorithm for circuit partitioning. Integration 30, 1–11 (2000) MATHGoogle Scholar
  14. 14.
    Nagamochi, H.: Algorithms for the minimum partitioning problems in graphs. IEICE Trans. Inf. Syst. J86-D-1, 53–68 (2003) Google Scholar
  15. 15.
    Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivity. Cambridge University Press, New York (2008) CrossRefMATHGoogle Scholar
  16. 16.
    Queyranne, M.: On optimum size-constrained set partitions. In: Proceedings of AUSSOIS 1999 (1999). http://www.iasi.cnr.it/iasi/aussois99/queyranne.html Google Scholar
  17. 17.
    Stoer, M., Wagner, F.: A simple min-cut algorithm. J. ACM 44, 585–591 (1997) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Thorup, M.: Minimum k-way cuts via deterministic greedy tree packing. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 159–166 (2008) Google Scholar
  19. 19.
    Vazirani, V.V., Yannakakis, M.: Suboptimal cuts: Their enumeration, weight and number. In: Proceedings of the 19th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 623, pp. 366–377. Springer, Berlin (1992) Google Scholar
  20. 20.
    Xiao, M.: Finding minimum 3-way cuts in hypergraphs. Inf. Process. Lett. 110, 554–558 (2010) CrossRefGoogle Scholar
  21. 21.
    Xiao, M.: An improved divide-and-conquer algorithm for finding all minimum k-way cuts. In: Proceedings of the 19th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 5369, pp. 208–219. Springer, Berlin (2008) Google Scholar
  22. 22.
    Zhao, L., Nagamochi, H., Ibaraki, T.: Greedy splitting algorithms for approximating multiway partition problems. Math. Program. 102, 167–183 (2005) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Kazumasa Okumoto
    • 1
  • Takuro Fukunaga
    • 2
  • Hiroshi Nagamochi
    • 2
  1. 1.Graduate School of EconomicsUniversity of TokyoTokyoJapan
  2. 2.Graduate School of InformaticsKyoto UniversityKyotoJapan

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