Abstract
Given an undirected graph G = (V,E) and three specified terminal nodes t 1,t 2,t 3, a 3-cut is a subset A of E such that no two terminals are in the same component of G\A. If a non-negative edge weight c e is specified for each e ∈ E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is NP-hard, and in fact, is max-SNP-hard. An approximation algorithm having performance guarantee 7/6 has recently been given by Călinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3-cut has weight at most 7/6 times the optimal LP value. It is proved here that 7/6 can be improved to 12/11, and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3-cut problem having performance guarantee 12/11.
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References
G. Călinescu, H. Karloff, and Y. Rabani: An improved approximation algorithm for MULTIWAY CUT Proceedings of Symposium on Theory of Computing, ACM, 1998.
Kevin Cheung, private communication, 1999.
S. Chopra and M.R. Rao, “On the multiway cut polyhedron”, Networks 21(1991), 51–89.
W.H. Cunningham, “The optimal multiterminal cut problem”, in: C. Monma and F. Hwang (eds.), Reliability of Computer and Communications Networks, American Math. Soc., 1991, pp. 105–120.
E. Dahlhaus, D. Johnson, C. Papadimitriou, P. Seymour, and M. Yannakakis, “The Complexity of multiway cuts”, extended abstract, 1983.
E. Dahlhaus, D. Johnson, C. Papadimitriou, P. Seymour, and M. Yannakakis, “The Complexity of multiterminal cuts”, SIAM J. Computing, 23(1994), 864–894.
D. Karger, P. Klein, C. Stein, M. Thorrup, and N. Young, “Rounding algorithms for a geometric embedding of minimum multiway cut,” Proceedings of Symposium on Theory of Computing, ACM, 1999, to appear.
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© 1999 Springer-Verlag Berlin Heidelberg
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Cunningham, W.H., Tang, L. (1999). Optimal 3-Terminal Cuts and Linear Programming. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_9
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DOI: https://doi.org/10.1007/3-540-48777-8_9
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