The dualization in arbitrary posets is a well-studied problem in combinatorial enumeration and is a crucial step in many applications in logics, databases, artificial intelligence and pattern mining.
The objective of this paper is to study reductions of the dualization problem on arbitrary posets to the dualization problem on boolean lattices, for which output quasi-polynomial time algorithms exist. Quasi-polynomial time algorithms are algorithms which run in where n is the size of the input and output. We introduce convex embedding and poset reflection as key notions to characterize such reductions. As a consequence, we identify posets, which are not boolean lattices, for which the dualization problem remains in quasi-polynomial time and propose a classification of posets with respect to dualization.
From these results, we study how they can be applied to maximal pattern mining problems. We deduce a new classification of pattern mining problems and we point out how known problems involving sequences and conjunctive queries patterns, fit into this classification. Finally, we explain how to adapt the seminal Dualize & Advance algorithm to deal with such patterns.
As far as we know, this is the first contribution to explicit non-trivial reductions for studying the hardness of maximal pattern mining problems and to extend the Dualize & Advance algorithm for complex patterns.
