On the complexity of monotone dualization and generating minimal hypergraph transversals

In 1994 Fredman and Khachiyan established the remarkable result that the duality of a pair of monotone Boolean functions, in disjunctive normal forms, can be tested in quasi-polynomial time, thus putting the problem, most likely, somewhere between polynomiality and coNP-completeness. We strengthen this result by showing that the duality testing problem can in fact be solved in polylogarithmic time using a quasi-polynomial number of processors (in the PRAM model). While our decomposition technique can be thought of as a generalization of that of Fredman and Khachiyan, it yields stronger bounds on the sequential complexity of the problem in the case when the sizes of f and g are significantly different, and also allows for generating all minimal transversals of a given hypergraph using only polynomial space. (C) 2007 Elsevier B.V. All rights reserved.