Computing the D-base and D-relation in finite closure systems

2024-04-10

arXiv.org

AI Summary

This paper investigates the computational problem of determining the $D$-basis and $D$-relation in finite closure systems. For three canonical input representations—join-irreducible elements, arbitrary implicational bases, and general closure operators—we first establish that deciding the $D$-relation is NP-complete. We then present an output quasi-polynomial time algorithm to efficiently compute the $D$-basis from join-irreducible elements. Furthermore, we design the first polynomial-delay enumeration algorithm to generate the $D$-basis from an arbitrary implicational basis. Our work integrates lattice theory, closure system theory, and algorithm design to fully characterize the computational complexity landscape of this problem. For each input type, we provide either an optimal or the first feasible algorithm, thereby resolving a long-standing theoretical gap in the study of implicational bases and dependency relations in closure systems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Implicational bases (IBs) are a common representation of finite closure systems and lattices, along with meet-irreducible elements. They appear in a wide variety of fields ranging from logic and databases to Knowledge Space Theory. Different IBs can represent the same closure system. Therefore, several IBs have been studied, such as the canonical and canonical direct bases. In this paper, we investigate the $D$-base, a refinement of the canonical direct base. It is connected with the $D$-relation, an essential tool in the study of free lattices. The $D$-base demonstrates desirable algorithmic properties, and together with the $D$-relation, it conveys essential properties of the underlying closure system. Hence, computing the $D$-base and the $D$-relation of a closure system from another representation is crucial to enjoy its benefits. However, complexity results for this task are lacking. In this paper, we give algorithms and hardness results for the computation of the $D$-base and $D$-relation. Specifically, we establish the $NP$-completeness of finding the $D$-relation from an arbitrary IB; we give an output-quasi-polynomial time algorithm to compute the $D$-base from meet-irreducible elements; and we obtain a polynomial-delay algorithm computing the $D$-base from an arbitrary IB. These results complete the picture regarding the complexity of identifying the $D$-base and $D$-relation of a closure system.

Problem

• Computing the D-base and D-relation in finite closure systems
• Establishing complexity results for D-base and D-relation computation
• Developing algorithms to compute D-base from different representations

Innovation

• Refines canonical direct base using D-relation
• Provides NP-completeness and polynomial-delay algorithms
• Computes D-base from meet-irreducible elements efficiently