Weighted Transversals of a Hypergraph

We consider a generalization of the notion of transversal to a finite hypergraph, so called weighted transversals. Given a non-negative weight vector assigned to each hyperedge of the input hypergraph, we define a weighted transversal as a minimal vertex set which intersects a collection of hyperedges of sufficiently large total weight. We show that the hypergraph of all weighted transversals is dual-bounded, i.e, the size of its dual 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. We also prove that the problem of generating all weighted transversals for a given hypergraph is polynomial-time reducible 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 case the generation of all the minimal Boolean solutions to a given system of non-negative linear inequalities.