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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 , 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.
