Abstract
We consider the problems of enumerating all minimal strongly connected subgraphs and all minimal dicuts of a given 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-complete 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.
This research was supported by the National Science Foundation (Grant IIS-0118635). The second and third authors are also grateful for the partial support by DIMACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical Computer Science.
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Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L. (2004). Enumerating Minimal Dicuts and Strongly Connected Subgraphs and Related Geometric Problems. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_12
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DOI: https://doi.org/10.1007/978-3-540-25960-2_12
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