Abstract
We investigate the bicriteria global minimum cut problem where each edge is evaluated by two nonnegative cost functions. The parametric complexity of such a problem is the number of linear segments in the parametric curve when we take all convex combinations of the criteria. We prove that the parametric complexity of the global minimum cut problem is O(|V|3). As a consequence, we show that the number of non-dominated points is O(|V|7) and give the first strongly polynomial time algorithm to compute these points. These results improve on significantly the super-polynomial bound on the parametric complexity given by Mulmuley [11], and the pseudo-polynomial time algorithm of Armon and Zwick [1] to solve this bicriteria problem. We extend some of these results to arbitrary cost functions and more than two criteria, and to global minimum cuts in hypergraphs.
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References
Armon, A., Zwick, U.: Multicriteria global minimum cuts. Algorithmica 46(1), 15–26 (2006)
Carstensen, P.: Complexity of some parametric integer and network programming problems. Mathematical Programming 26, 64–75 (1983)
Ehrgott, M.: Multicriteria Optimization. Springer (2005)
Henzinger, M., Williamson, D.P.: On the number of small cuts in a graph. Information Processing Letters 59(1), 41–44 (1996)
Ihler, E., Wagner, D., Wagner, F.: Modeling hypergraphs by graphs with the same mincut properties. Information Processing Letters 45(4), 171–175 (1993)
Karger, D.R.: Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. In: Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 21–30 (1993)
Karger, D.R.: Minimum cuts in near-linear time. Journal of the ACM 47(1), 46–76 (2000)
Karger, D.R., Stein, C.: A new approach to the minimum cut problem. Journal of the ACM 43(4), 601–640 (1996)
Klimmek, R., Wagner, F.: A simple hypergraph min cut algorithm. Freie Univ., Fachbereich Mathematik (1996)
McCormick, S.T., Ervolina, T.R.: Computing maximum mean cuts. Discrete Applied Math 52, 53–70 (1994)
Mulmuley, K.: Lower bounds in a parallel model without bit operations. SIAM Journal on Computing 28(4), 1460–1509 (1999)
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.: Algorithmic aspects of graph connectivity. Cambridge University Press (2008)
Nagamochi, H., Nakamura, S., Ishii, T.: Constructing a cactus for minimum cuts of a graph in O(mn + n 2 logn) time and O(m) space. Inst. Electron. Inform. Comm. Eng. Trans. Inform. Systems, 179–185 (2003)
Nagamochi, H., Nishimura, K., Ibaraki, T.: Computing all small cuts in undirected networks. SIAM Journal of Discrete Mathematics 10, 469–481 (1997)
Queyranne, M.: Minimizing symmetric submodular functions. Mathematical Programming 82(1-2), 3–12 (1998)
Radzik, T.: Newton’s Method for Fractional Combinatorial Optimization. In: Proceedings of IEEE Annual Symp. of Foundations of Computer Science, pp. 659–669 (1992)
Stoer, M., Wagner, F.: A simple min-cut algorithm. Journal of the ACM 44(4), 585–591 (1997)
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Aissi, H., Mahjoub, A.R., McCormick, S.T., Queyranne, M. (2014). A Strongly Polynomial Time Algorithm for Multicriteria Global Minimum Cuts. In: Lee, J., Vygen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2014. Lecture Notes in Computer Science, vol 8494. Springer, Cham. https://doi.org/10.1007/978-3-319-07557-0_3
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DOI: https://doi.org/10.1007/978-3-319-07557-0_3
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