Dual-Bounded Hypergraphs: A Survey∗

This short paper surveys recent results on the generation of implicitly given hypergraphs, as well as their applications in data mining, reliability theory, integer programming and combinatorics. More precisely, we consider a monotone property π over the subsets of a finite set V , the corresponding family Sπ of subsets satisfying property π, and the problem of generating (sequentially) the family Fπ of all minimal subsets in Sπ, when only V is given explicitly, and π is represented by an oracle Oπ. We show that for a number of interesting monotone properties, the family Fπ is uniformly dual-bounded allowing for the incrementally efficient generation 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 maximal subsets not having property π is NP-hard for most of the monotone properties π considered in this paper. ∗The research of the first four authors was supported in part by the National Science Foundation Grant IIS-0118635. 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. †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). §Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan; ([email protected]).