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.
Similar content being viewed by others
References
Frank, A.: Applications of submodular functions. In: Surveys in Combinatorics. Cambridge London Mathematical Society Lecture Notes Series, vol. 187, pp. 85–136 (1993)
Fujishige, S.: Submodular Function and Optimization. North-Holland, Amsterdam (1991)
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)
Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. J. Algorithms 50, 49–61 (2004)
Gasieniec, L., Jansson, J., Lingas, A., Óstlin, A.: On the complexity of constructing evolutionary trees. J. Comb. Optim. 3, 183–197 (1999)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. ACM 35, 921–940 (1988)
Goldschmidt, O., Hochbaum, D.: A polynomial algorithm for the k-cut problem for fixed k. Math. Oper. Res. 19, 24–37 (1994)
Iwata, S.: Submodular function minimization. Math. Program. 112, 45–64 (2008)
Kamidoi, Y., Yoshida, N., Nagamochi, H.: A deterministic algorithm for finding all minimum k-way cuts. SIAM J. Comput. 36, 1329–1341 (2006)
Karger, D.R., Stein, C.: A new approach to the minimum cut problem. J. ACM 43, 601–640 (1996)
Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Internal Report B 96-02, Bericht FU Berlin Fachbereich Mathematik und Informatik (1995)
Lawler, E.L.: Cutsets and partitions of hypergraphs. Networks 3, 275–285 (1973)
Mak, W.-K., Wong, D.F.: A fast hypergraph min-cut algorithm for circuit partitioning. Integration 30, 1–11 (2000)
Nagamochi, H.: Algorithms for the minimum partitioning problems in graphs. IEICE Trans. Inf. Syst. J86-D-1, 53–68 (2003)
Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivity. Cambridge University Press, New York (2008)
Queyranne, M.: On optimum size-constrained set partitions. In: Proceedings of AUSSOIS 1999 (1999). http://www.iasi.cnr.it/iasi/aussois99/queyranne.html
Stoer, M., Wagner, F.: A simple min-cut algorithm. J. ACM 44, 585–591 (1997)
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)
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)
Xiao, M.: Finding minimum 3-way cuts in hypergraphs. Inf. Process. Lett. 110, 554–558 (2010)
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)
Zhao, L., Nagamochi, H., Ibaraki, T.: Greedy splitting algorithms for approximating multiway partition problems. Math. Program. 102, 167–183 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-010-9483-0