A Parallel Approach to Counting Exact Covers Based on Decomposability Property

TL;DR

This paper introduces a new parallel algorithm, DXD, that constructs a decision-ZDNNF for counting exact covers more efficiently than existing methods, leveraging a novel decomposability property.

Contribution

It proposes a decision-ZDNNF representation and a parallel algorithm DXD for counting exact covers, improving efficiency over prior approaches.

Findings

DXD outperforms state-of-the-art methods in experiments.

Decision-ZDNNF is more succinct than ZBDDs.

Dynamic updates of connected components enhance performance.

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.