A Parallel Approach to Counting Exact Covers Based on Decomposability Property

2026-04-16

AI Summary

Exact cover counting is a classic NP-hard problem in artificial intelligence. This work proposes DXD, a parallel algorithm leveraging a decomposability property, whose key innovation lies in introducing decision-ZDNNF (Zero-suppressed Decision Decomposable Negation Normal Form)—a more compact knowledge representation than ZBDDs—and designing the first parallel counting algorithm based on this structure. The approach further integrates a dynamic connected-component updating strategy to enhance computational efficiency. Experimental results demonstrate that DXD significantly outperforms state-of-the-art methods across multiple benchmark instances.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The exact cover problem is a classical NP-hard problem with broad applications in the area of AI. Algorithm DXZ is a method to count exact covers representing by zero-suppressed binary decision diagrams (ZBDDs). In this paper, we propose a zero-suppressed variant of decision decomposable negation normal form (in short, decision-ZDNNF), which is strictly more succinct than ZBDDs. We then design a novel parallel algorithm, namely DXD, which constructs a decision-ZDNNF representing the set of all exact covers. Furthermore, we improve DXD by dynamically updating connected components. The experimental results demonstrate that the improved DXD algorithm outperforms all of state-of-the-art methods.

Problem

• exact cover
• counting
• NP-hard
• AI
• knowledge representation

Innovation

• decision-ZDNNF
• exact cover counting
• parallel algorithm
• knowledge compilation
• dynamic connected components