Scientific contributions of Leo Khachiyan (a short overview)
Discrete Applied Mathematics
Lower bounds for three algorithms for transversal hypergraph generation
Discrete Applied Mathematics
An incremental polynomial time algorithm to enumerate all minimal edge dominating sets
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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.