Generating Partial and Multiple Transversals of a Hypergraph

We consider two natural generalizations of the notion of transversal to a finite hypergraph, arising in data-mining and machine learning, the so called multiple and partial transversals. We show that the hypergraphs of all multiple and all partial transversals are dualbounded in the sense that in both cases, the size of the dual hypergraph is bounded by a polynomial in the cardinality and the length of description of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold logic, which may be of independent interest. We also show that the problems of generating all multiple and all partial transversals of an arbitrary hypergraph are polynomial-time reducible to the well-known dualization problem of hypergraphs. As a corollary, we obtain incremental quasi-polynomial-time algorithms for both of the above problems, as well as for the generation of all the minimal Boolean solutions for an arbitrary monotone system of linear inequalities. Thus, it is unlikely that these problems are NP-hard.