Enumerating Minimal Dicuts and Strongly Connected Subgraphs and Related Geometric Problems

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 A Rn , it is NP-hard to generate all maximal subsets of A contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of 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.

• Type: Conference
• Year: 2004
• Where: IPCO
• Authors: Endre Boros, Khaled M. Elbassioni, Vladimir Gurvich, Leonid Khachiyan
• Keywords: Incremental Polynomial Time, IPCO 2004, IPCO 2007, Minimal Dicuts, Minimal Subsets