New Algorithms for #2-SAT and #3-SAT

2025-07-19

AI Summary

This paper addresses the exact model counting problem for #2-SAT and #3-SAT—i.e., computing the total number of satisfying assignments to a Boolean formula. To overcome the high time complexity of classical algorithms, we introduce three novel techniques: (1) new local reduction rules that substantially shrink formula size; (2) a path-decomposition framework leveraging both primal and dual graphs to enable superior branching strategies; and (3) a synergistic mechanism integrating weighted model counting with fine-grained backtracking during branching. Theoretical analysis shows our algorithm achieves time complexities of $O^*(1.1082^m)$ for #2-SAT and $O^*(1.4423^m)$ for #3-SAT, improving upon the prior best bound of $O^*(1.1892^m)$. These results establish new upper bounds on the computational complexity of these fundamental #SAT problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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 #2-SAT and #3-SAT problems involve counting the number of satisfying assignments (also called models) for instances of 2-SAT and 3-SAT, respectively. In 2010, Zhou et al. proposed an $mathcal{O}^*(1.1892^m)$-time algorithm for #2-SAT and an efficient approach for #3-SAT, where $m$ denotes the number of clauses. In this paper, we show that the weighted versions of #2-SAT and #3-SAT can be solved in $mathcal{O}^*(1.1082^m)$ and $mathcal{O}^*(1.4423^m)$ time, respectively. These results directly apply to the unweighted cases and achieve substantial improvements over the previous results. These advancements are enabled by the introduction of novel reduction rules, a refined analysis of branching operations, and the application of path decompositions on the primal and dual graphs of the formula.

Problem

• Develop faster algorithms for weighted #2-SAT and #3-SAT problems
• Improve time complexity to O*(1.1082^m) for #2-SAT
• Enhance efficiency using reduction rules and path decompositions

Innovation

• Novel reduction rules for weighted SAT
• Refined branching operations analysis
• Path decompositions on primal graphs