Translating between the representations of an acyclic convex geometry of bounded degree
Abstract
We consider the problem of enumerating the irreducible closed sets of a closure system given by an implicational base. In the context of Horn logic, these correspond to Horn expressions and characteristic models, respectively. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph dualization problem, even in the context of acyclic convex geometries, i.e., closure systems admitting an acyclic implicational base. This paper studies this later class with a focus on the degree, which corresponds to the maximal number of implications in which an element occurs. We show that the problem is tractable for bounded values of this parameter, even when relaxed to the notions of premise- and conclusion-degree. Our algorithms rely on structural properties of acyclic convex geometries and involve various techniques from algorithmic enumeration such as solution graph traversal, saturation techniques, and a sequential approach leveraging from acyclicity. They are shown to perform in incremental-polynomial time for the computation of irreducible closed sets, and in polynomial time for the construction of an implicational base. Finally, we argue that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.