Skip to main content
SpringerLink
Log in

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

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The submodular system k-partition problem is a problem of partitioning a given finite set V into k non-empty subsets V 1,V 2,…,V k 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Frank, A.: Applications of submodular functions. In: Surveys in Combinatorics. Cambridge London Mathematical Society Lecture Notes Series, vol. 187, pp. 85–136 (1993)

    Chapter  Google Scholar 

  2. Fujishige, S.: Submodular Function and Optimization. North-Holland, Amsterdam (1991)

    Google Scholar 

  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. Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. J. Algorithms 50, 49–61 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gasieniec, L., Jansson, J., Lingas, A., Óstlin, A.: On the complexity of constructing evolutionary trees. J. Comb. Optim. 3, 183–197 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. ACM 35, 921–940 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Goldschmidt, O., Hochbaum, D.: A polynomial algorithm for the k-cut problem for fixed k. Math. Oper. Res. 19, 24–37 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Iwata, S.: Submodular function minimization. Math. Program. 112, 45–64 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kamidoi, Y., Yoshida, N., Nagamochi, H.: A deterministic algorithm for finding all minimum k-way cuts. SIAM J. Comput. 36, 1329–1341 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Karger, D.R., Stein, C.: A new approach to the minimum cut problem. J. ACM 43, 601–640 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Internal Report B 96-02, Bericht FU Berlin Fachbereich Mathematik und Informatik (1995)

  12. Lawler, E.L.: Cutsets and partitions of hypergraphs. Networks 3, 275–285 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mak, W.-K., Wong, D.F.: A fast hypergraph min-cut algorithm for circuit partitioning. Integration 30, 1–11 (2000)

    MATH  Google Scholar 

  14. Nagamochi, H.: Algorithms for the minimum partitioning problems in graphs. IEICE Trans. Inf. Syst. J86-D-1, 53–68 (2003)

    Google Scholar 

  15. Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivity. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  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. Stoer, M., Wagner, F.: A simple min-cut algorithm. J. ACM 44, 585–591 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Xiao, M.: Finding minimum 3-way cuts in hypergraphs. Inf. Process. Lett. 110, 554–558 (2010)

    Article  Google Scholar 

  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. Zhao, L., Nagamochi, H., Ibaraki, T.: Greedy splitting algorithms for approximating multiway partition problems. Math. Program. 102, 167–183 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Economics, University of Tokyo, Tokyo, 113-0033, Japan

    Kazumasa Okumoto

  2. Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Takuro Fukunaga & Hiroshi Nagamochi

Authors
  1. Kazumasa Okumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takuro Fukunaga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hiroshi Nagamochi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takuro Fukunaga.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Okumoto, K., Fukunaga, T. & Nagamochi, H. Divide-and-Conquer Algorithms for Partitioning Hypergraphs and Submodular Systems. Algorithmica 62, 787–806 (2012). https://doi.org/10.1007/s00453-010-9483-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-010-9483-0

Keywords

Search

Navigation