Generating dual-bounded hypergraphs

This article surveys some recent results on the generation of implicitly given hypergraphs and their applications in Boolean and integer programming, data mining, reliability theory, and combinatorics. Given a monotone property pi over the subsets of a finite set V , we consider the problem of incrementally generating the family F of all minimal subsets satisfying property pi, when pi is given by a polynomial-time satisfiability oracle. For a number of interesting monotone properties, the family F-pi turns out to be uniformly dual-bounded, allowing for the incrementally efficient enumeration of the members of F-pi. Important applications include the efficient generation of minimal infrequent sets of a database (data mining), minimal connectivity ensuring collections of subgraphs from a given list (reliability theory), minimal feasible solutions to a system of monotone inequalities in integer variables (integer programming), minimal spanning collections of subspaces from a given list (linear algebra) and maximal independent sets in the intersection of matroids (combinatorial optimization). In contrast to these results, the analogous problem of generating the family of all maximal subsets not having property pi is NP-hard for almost all applications mentioned above.