On Dualization over Distributive Lattices
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Given a partially order set (poset) P, and a pair of families of ideals and filters in P such that each pair (I,F)∈× has a non-empty intersection, the dualization problem over P is to check whether there is an ideal X in P which intersects every member of and does not contain any member of . Equivalently, the problem is to check for a distributive lattice L=L(P), given by the poset P of its set of joint-irreducibles, and two given antichains ,⊆ L such that no a∈ is dominated by any b∈, whether and cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of P, and , thus answering an open question in <cit.>. As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.
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