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, and k is a fixed integer. This problem contains the hypergraph k-cut problem. In this paper, we design the first exact algorithm for k = 3 and approximation algorithms for k ≥ 4. We also analyze the approximation factor for the hypergraph k-cut problem.
This work was partially supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Frank, A.: Applications of Submodular Functions. 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. Technical report, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University (2009)
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. Opt. 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. Operations Research 19, 24–37 (1994)
Iwata, S.: Submodular Function Minimization. Math. Prog. 112, 45–64 (2008)
Kamidoi, Y., Yoshida, N., Nagamochi, H.: A Deterministic Algorithm for Finding All Minimum k-way Cuts. SIAM J. Comp. 36, 1329–1341 (2006)
Karger, D.R., Stein, C.: A New Approach to the Minimum Cut Problem. J. ACM 43, 601–640 (1996)
Lawler, E.L.: Cutsets and Partitions of Hypergraphs. Networks 3, 275–285 (1973)
Nagamochi, H.: Algorithms for the Minimum Partitioning Problems in Graphs. IEICE Trans. Inf. Sys. J86-D-1, 53–68 (2003)
Queyranne, M.: On Optimum Size-constrained Set Partitions. In: Proceedings of AUSSOIS (1999)
Thorup, M.: Minimum k-way Cuts via Deterministic Greedy Tree Packing. In: 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: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 366–377. Springer, Heidelberg (1992)
Xiao, M.: Finding Minimum 3-way Cuts in Hypergraphs. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 270–281. Springer, Heidelberg (2008)
Xiao, M.: An Improved Divide-and-Conquer Algorithm for Finding All Minimum k-Way Cuts. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 208–219. Springer, Heidelberg (2008)
Zhao, L., Nagamochi, H., Ibaraki, T.: A Unified Framework for Approximating Multiway Partition Problems. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 682–694. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Okumoto, K., Fukunaga, T., Nagamochi, H. (2009). Divide-and-Conquer Algorithms for Partitioning Hypergraphs and Submodular Systems. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-10631-6_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10630-9
Online ISBN: 978-3-642-10631-6
eBook Packages: Computer ScienceComputer Science (R0)