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 \(\mathcal{A}\subseteq\mathbb{R}^{n}\) , it is NP-hard to generate all maximal subsets of \(\mathcal{A}\) contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of \(\mathcal{A}\) 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.
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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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Khachiyan, L., Boros, E., Elbassioni, K. et al. On Enumerating Minimal Dicuts and Strongly Connected Subgraphs. Algorithmica 50, 159–172 (2008). https://doi.org/10.1007/s00453-007-9074-x
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DOI: https://doi.org/10.1007/s00453-007-9074-x