On Dualization over Distributive Lattices [article]

Khaled Elbassioni
2022 arXiv   pre-print
Given a partially order set (poset) P, and a pair of families of ideals ℐ and filters ℱ in P such that each pair (I,F)∈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 ℱ and does not contain any member of ℐ. 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 𝒜,ℬ⊆ L such that no a∈𝒜 is dominated by any
more » ... ∈ℬ, whether 𝒜 and ℬ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of P, 𝒜 and ℬ, thus answering an open question in Babin and Kuznetsov (2017). 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.
arXiv:2006.15337v4 fatcat:2eqgc3ldmnfbdkzbjoy2pv7dxu