Abstract
The minimum 3-way cut problem in an edge-weighted hypergraph is to find a partition of the vertices into 3 sets minimizing the total weight of hyperedges with at least two endpoints in two different sets. In this paper we present some structural properties for minimum 3-way cuts and design an O(dmn 3) algorithm for the minimum 3-way cut problem in hypergraphs, where n and m are the numbers of vertices and edges respectively, and d is the sum of the degrees of all the vertices. Our algorithm is the first deterministic algorithm finding minimum 3-way cuts in hypergraphs.
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References
Burlet, M., Goldschmidt, O.: A new and improved algorithm for the 3-cut problem. Operations Research Letters 21(5), 225–227 (1997)
Easley, R.F., Hartvigsen, D.: Crossing properties of multiterminal cuts. Networks 34(3), 215–220 (1999)
Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. J. ACM. 45(5), 783–797 (1998): A preliminary version appeared in FOCS 1997
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)
Goldschmidt, O., Hochbaum, D.: A polynomial algorithm for the k-cut problem for fixed k. Mathematics of Operations Research 19(1), 24–37 (1994): A preliminary version appeared in FOCS 1988
Gomory, R.E., Hu, T.C.: Multi-terminal network flows. J. SIAM. 9(4), 551–670 (1961)
Kamidoi, Y., Wakabayashi, S., Yoshida, N.: A divide-and-conquer approach to the minimum k-way cut problem. Algorithmica 32(2), 262–276 (2002)
Kamidoi, Y., Yoshida, N., Nagamochi, H.: A deterministic algorithm for finding all minimum k-way cuts. SIAM Journal on Computing 36(5), 1329–1341 (2006)
Kapoor, S.: On minimum 3-cuts and approximating k-cuts using cut trees. In: Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization, Springer, London (1996)
Karger, D.R., Stein, C.: A new approach to the minimum cut problem. Journal of the ACM 43(4), 601–640 (1996): Preliminary portions appeared in SODA 1993 and STOC 1993
Karypis, G., Kumar, V.: hmetis: A hypergraph partitioning package version 1.5, user manual (1998), http://glaros.dtc.umn.edu/gkhome/fetch/sw/hmetis/manual.pdf
Karypis, G., Kumar, V.: Multilevel k-way hypergraph partitioning. VLSI Design 11(3), 285–300 (2000)
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(3), 275–285 (1973)
Levine, M.S.: Fast randomized algorithms for computing minimum {3,4,5,6}-way cuts. In: Proceedings of the 11th annual ACM-SIAM symposium on Discrete algorithms (SODA 2000), Philadelphia, PA, USA. Society for Industrial and Applied Mathematics (2000)
Mak, W.-K., Wong, D.F.: A fast hypergraph min-cut algorithm for circuit partitioning. Integration, the VLSI Journal 30(1), 1–11 (2000)
Nagamochi, H., Ibaraki, T.: Computing edge connectivity in multigraphs and capacitated graphs. SIAM Journal on Discrete Mathematics 5(1), 54–66 (1992)
Nagamochi, H., Ibaraki, T.: A fast algorithm for computing minimum 3-way and 4-way cuts. Mathematical Programming 88(3), 507–520 (2000)
Nagamochi, H., Katayama, S., Ibaraki, T.: A Faster Algorithm for Computing Minimum 5-Way and 6-Way Cuts in Graphs. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, Springer, Heidelberg (1999)
Preas, B.T., Lorenzetti, M.: Physical Design Automation of VLSI Systems, Benjamin-Cummings, California (1988)
Stoer, M., Wagner, F.: A simple min-cut algorithm. Journal of the ACM 44(4), 585–591 (1997); A preliminary version appeared in ESA 1994
Vazirani, V.V., Yannakakis, M.: Suboptimal cuts: Their enumeration, weight and number. In: Kuich, W. (ed.) Automata, Languages and Programming. Proc. of the 19th International Colloquium, Springer, Berlin (1992)
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Xiao, M. (2008). Finding Minimum 3-Way Cuts in Hypergraphs. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_24
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DOI: https://doi.org/10.1007/978-3-540-79228-4_24
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