On the dualization in distributive lattices and related problems

TL;DR

This paper explores the dualization problem in distributive lattices, providing new formulations and analyzing complexity across various graph and poset classes, revealing both tractable and intractable cases.

Contribution

It introduces equivalent formulations of dualization in distributive lattices using graphs, hypergraphs, and posets, and studies the problem's complexity under different restrictions.

Findings

Enumeration of minimal dominating sets is linear delay in split graphs.

Dualization becomes as hard as in general graphs under certain poset restrictions.

Some restricted cases allow for polynomial-time dualization algorithms.

Abstract

In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem. We in particular propose equivalent formulations of the problem in terms of graphs, hypergraphs, and posets. It is known that hypergraph dualization amounts to generate all minimal transversals of a hypergraph, or all minimal dominating sets of a graph. In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal ``ideal solutions'' are to be generated. This in particular allows us to study the complexity of the problem under various combined restrictions on graph classes and poset types, including bipartite, split, and co-bipartite graphs, and variants of neighborhood inclusion posets. We for example show that while the enumeration of minimal dominating sets is possible with linear delay in split graphs, the problem, within the same class, gets as hard as for general graphs when generalized to this framework. More surprisingly, this result holds even when the poset is only comparing vertices of included neighborhoods in the graph. If both the poset and the graph class are sufficiently restricted, we show that the dualization is tractable relying on existing algorithms from the literature.