Dualization in lattices given by implicational bases

Abstract

It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P = NP. In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. In the case of premises of size one—when the lattice is distributive—we show that the dualization is possible in output quasi-polynomial time whenever the graph of implications is of bounded maximum induced matching. Lattices that share this property include distributive lattices coded by the ideals of an interval order.