# Dualization in lattices given by implicational bases

## TL;DR

This paper investigates the computational complexity of dualization in lattices defined by implicational bases, showing hardness results and identifying cases where quasi-polynomial algorithms are possible based on hypergraph dualization techniques.

## Contribution

It proves that dualization remains hard even with small premises and introduces a quasi-polynomial algorithm for bases with bounded independent-width.

## Key findings

- Dualization is NP-hard even with premises of size at most two.
- A quasi-polynomial time algorithm exists for bases with bounded independent-width.
- Distributive lattices from interval orders have bounded independent-width and are efficiently dualizable.

## 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. Then we show using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent conclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one.