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

2025-06-30

AI Summary

This paper addresses the enumeration of irredundant closed sets generated by an implication base in acyclic convex geometries of bounded degree—a problem equivalent to enumerating characteristic models in Horn logic and generalizing hypergraph dualization, whose complexity has long remained open. The authors introduce the novel concept of “premise-conclusion degree relaxation,” revealing that the synergy of acyclicity and bounded degree enables circumvention of the hypergraph dualization bottleneck. Leveraging solution-graph traversal, saturation techniques, and serialization of acyclic structures, they devise an incremental polynomial-time algorithm for enumerating irredundant closed sets and constructing the implication base in polynomial time. Theoretically, they prove that flashlight search cannot achieve polynomial delay, thereby establishing the necessity and optimality of their approach.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Enumerating irreducible closed sets in closure systems
• Analyzing complexity in acyclic convex geometries
• Developing efficient algorithms for bounded degree cases

Innovation

• Acyclic convex geometries with bounded degree
• Algorithmic enumeration techniques
• Incremental-polynomial time computation