Abstract
Let λ(N) denote the weight of a minimum cut in an edge-weighted undirected network N, where n and m are the numbers of vertices and edges, respectively. It is known that O(n 2k) is an upper bound on the number of cuts with weights less than kλ(N). We first show that all cuts of weights less than kλ(N) can be enumerated in O(mn 3 + n 2k+2) time without using the maximum flow algorithm. We then prove for k<4/3 that ( n2 ) is a tight upper bound on the number of cuts of weights less than kλ(N).
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References
R. Bixby, The minimum number of edges and vertices in a graph with edge-connectivity n and m n-bonds, Networks, Vol.5, 1975, pp. 235–298.
C.J. Colbourn, The combinatorics of network reliability, Oxford University Press, New York Oxford, 1987.
E. Dinits, A.V. Karzanov and M.V. Lomonosov, On the structure of a family of minimal weighed cuts in a graph, Studies in Discrete Optimization (in Russian), A. A. Fridman (Ed.), Nauka, Moscow, 1976, pp. 290–306.
A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Mathematics, Vol.5, 1992, pp. 25–53.
A. Frank, On a theorem of Mader, Discrete Mathematics, Vol.101, 1992, pp. 49–57.
D.R. Karger, Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm, Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms, 1993, pp. 21–30.
L. Lovász, Combinatorial Problems and Exercises, North-Holland 1979.
W. Mader, A reduction method for edge-connectivity in graphs, Annals of Discrete Mathematics, Vol. 3, 1978, pp. 145–164.
H. Nagamochi and T. Ibaraki, Computing the edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Mathematics, Vol 5, 1992, pp. 54–66.
H. Nagamochi, K. Nishimura and T. Ibaraki, Computing all small cuts in undirected networks, Technical Report #94007, Kyoto University, 1994.
D. Naor, D. Gusfield and C. Martel, A fast algorithm for optimally increasing the edge connectivity, Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science, 1990, pp. 698–707.
V.V. Vazirani and M. Yannakakis, Suboptimal cuts: Their enumeration, weight, and number, Lecture Notes in Computer Science, Vol. 623, Springer-Verlag, 1992, pp. 366–377.
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© 1994 Springer-Verlag Berlin Heidelberg
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Nagamochi, H., Nishimura, K., Ibaraki, T. (1994). Computing all small cuts in undirected networks. In: Du, DZ., Zhang, XS. (eds) Algorithms and Computation. ISAAC 1994. Lecture Notes in Computer Science, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58325-4_181
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DOI: https://doi.org/10.1007/3-540-58325-4_181
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