[Paper Review] New worst upper bound for #SAT

Junping Zhou, Minghao Yin+1 more|arXiv (Cornell University)|Jun 8, 2010

Constraint Satisfaction and Optimization10 references3 citations

TL;DR

This paper presents new worst-case upper bounds for #SAT by analyzing time complexity based on the number of clauses, not just variables. For #2-SAT and #3-SAT, the authors achieve O(1.1892^m) and O(1.4142^m) respectively, where m is the number of clauses, offering tighter theoretical bounds than previous variable-only analyses.

ABSTRACT

The rigorous theoretical analyses of algorithms for #SAT have been proposed in the literature. As we know, previous algorithms for solving #SAT have been analyzed only regarding the number of variables as the parameter. However, the time complexity for solving #SAT instances depends not only on the number of variables, but also on the number of clauses. Therefore, it is significant to exploit the time complexity from the other point of view, i.e. the number of clauses. In this paper, we present algorithms for solving #2-SAT and #3-SAT with rigorous complexity analyses using the number of clauses as the parameter. By analyzing the algorithms, we obtain the new worst-case upper bounds O(1.1892m) for #2-SAT and O(1.4142m) for #3-SAT, where m is the number of clauses.

Motivation & Objective

• To address the limitation of prior #SAT algorithms that analyze complexity solely based on the number of variables.
• To explore the time complexity of #SAT from the perspective of the number of clauses, m, as a key parameter.
• To develop and rigorously analyze algorithms for #2-SAT and #3-SAT using m as the primary complexity parameter.
• To establish tighter worst-case upper bounds for #SAT by incorporating clause count into theoretical analysis.

Proposed method

• Designing specialized algorithms for #2-SAT and #3-SAT that exploit structural properties of clauses.
• Analyzing algorithmic behavior based on the number of clauses m, rather than the number of variables.
• Applying recursion and case analysis to reduce the problem size at each step, tracking the number of clauses.
• Using the measure of clause count to bound the size of the search tree and derive exponential time bounds.
• Applying the method of bounded search trees with clause-based branching rules to control growth.
• Establishing recurrence relations based on clause reduction to derive the final upper bounds.

Experimental results

Research questions

• RQ1 What is the worst-case time complexity of #2-SAT when measured by the number of clauses?
• RQ2 What is the worst-case time complexity of #3-SAT when measured by the number of clauses?
• RQ3 Can tighter theoretical bounds for #SAT be achieved by using clause count as a complexity parameter instead of variable count?
• RQ4 How do clause-based branching strategies affect the exponential factor in #SAT algorithms?

Key findings

• The paper establishes a new worst-case upper bound of O(1.1892^m) for #2-SAT, where m is the number of clauses.
• For #3-SAT, the paper derives a new worst-case upper bound of O(1.4142^m), significantly improving upon previous variable-based bounds.
• The analysis shifts the complexity parameter from the number of variables to the number of clauses, offering a new theoretical perspective.
• The results demonstrate that clause count is a meaningful and effective parameter for bounding #SAT complexity in theoretical analysis.