Translating between the representations of an acyclic convex geometry of bounded degree

TL;DR

This paper investigates the complexity of translating between different representations in acyclic convex geometries, demonstrating tractability for bounded degree cases and establishing limits on algorithmic improvements.

Contribution

It introduces algorithms for translation problems in acyclic convex geometries with bounded degree, utilizing structural properties and enumeration techniques.

Findings

Problem is tractable for bounded degree values.

Algorithms operate in incremental-polynomial time.

Running times cannot be improved to polynomial delay.

Abstract

We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. 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. Finally, we complete these results by showing that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.