Dual-bounded generating problems: weighted transversals of a hypergraph

Abstract

We consider a generalization of the notion of transversal to a finite hypergraph, the so-called weighted transversals. Given a non-negative weight vector assigned to each hyperedge of an input hypergraph A and a non-negative threshold vector, we define a weighted transversal as a minimal vertex set which intersects all the hyperedges of A except for a sub-family of total weight not exceeding the given threshold vector. Weighted transversals generalize partial and multiple transversals introduced in Boros et al. (SIAM J. Comput. 30 (6) (2001)) and also include minimal binary solutions to non-negative systems of linear inequalities and minimal weighted infrequent sets in databases. We show that the hypergraph of all weighted transversals is dual-bounded, i.e., the size of its transversal hypergraph is polynomial in the number of weighted transversals and the size of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold Boolean logic, which may be of independent interest. For instance, we show that for any row-weighted m× n binary matrix and any threshold weight t, the number of maximal sets of columns whose row support has weight above t is at most m times the number of minimal sets of columns with row support of total weight below t. We also prove that the problem of generating all weighted transversals for a given hypergraph is polynomial-time reducible to the generation of all ordinary transversals for another hypergraph, i.e., to the well-known hypergraph dualization problem. As a corollary, we obtain an incremental quasi-polynomial-time algorithm for generating all weighted transversals for a given hypergraph. This result includes as special cases the generation of all the minimal Boolean solutions to a given system of non-negative linear inequalities and the generation of all minimal weighted infrequent sets of columns for a given binary matrix.