On Enumerating Minimal Dicuts and Strongly Connected Subgraphs |
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Authors: | Leonid Khachiyan Endre Boros Khaled Elbassioni Vladimir Gurvich |
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Affiliation: | (1) RUTCOR, Rutgers University, 640 Bartholomew Road, 08854-8003 Piscataway, NJ, USA;(2) Max-Planck-Institut für Informatik, Saarbrücken, Germany |
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Abstract: | We consider the problems of enumerating all minimal strongly connected subgraphs and all minimal dicuts of a given strongly
connected directed graph G=(V,E). We show that the first of these problems can be solved in incremental polynomial time, while the second problem is NP-hard:
given a collection of minimal dicuts for G, it is NP-hard to tell whether it can be extended. The latter result implies, in particular, that for a given set of points
, it is NP-hard to generate all maximal subsets of
contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of
whose convex hull contains the origin as an interior point, and show that this problem includes as a special case the well-known
hypergraph transversal problem.
This research was supported by the National Science Foundation (Grant IIS-0118635). The third and fourth authors are also
grateful for the partial support by DIMACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical
Computer Science.
Our friend and co-author, Leonid Khachiyan tragically passed away on April 29, 2005. |
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