Generating dual-bounded hypergraphs

This paper 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 π over the subsets of a finite set V , we consider the problem of incrementally generating the family Fπ of all minimal subsets satisfying property π, when π is given by a polynomialtime satisfiability oracle. For a number of interesting monotone properties, the family Fπ turns out to be uniformly dual-bounded, allowing for the incrementally efficient enumeration of the members of Fπ. 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 π is NP-hard for most of the monotone properties π considered in the paper. †RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003; ({boros,gurvich}@rutcor.rutgers.edu). ‡Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854-8003; ({elbassio@paul, leonid@cs}.rutgers.edu). ∗The research was supported in part by the National Science Foundation Grant IIS0118635. The research of the first and third authors was also supported in part by the Office of Naval Research Grant N00014-92-J-1375. 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.