A Hypergraph Container Method on Spread SAT: Approximation and Speedup

2026-04-16

AI Summary

This work investigates how the uniformity of clause distribution—captured by the “spread” property—can be leveraged to enhance the efficiency of Boolean satisfiability (SAT) solving and to effectively distinguish unsatisfiable formulas from those satisfying at least a $(1-\delta)$-fraction of clauses. We introduce a novel combination of hypergraph container methods with weighted $(\lambda,p)_k$-structures to characterize clause distributions in non-uniform SAT instances. Exploiting the near-independence properties of containers, we achieve a subexponential-time algorithm for Gap-SAT. Our results extend Zamir’s (STOC 2023) framework to the non-uniform setting, revealing that the parameter $\lambda$ directly governs the algorithmic speedup. Consequently, highly spread formulas are shown not to represent worst-case instances under Gap-ETH, leading to faster exact SAT solvers.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Distributed Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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

We develop a hypergraph container method for the Boolean Satisfiability Problem (SAT) via the newly developed container results [Campos and Samotij (2024)]. This provides an explicit connection between the extent of spread of clauses and the efficiency of container-based algorithms. Informally, the more evenly the clauses are distributed, the stronger the shrinking effect of the containers, which leads to faster algorithms for SAT. To quantify the extent of spread, we use a weighted point of view, in which a clause of size $s$ receives weight $p^s$ for some $0<p\le 1$.In this way, we introduce the notion of $(λ,p)_k$-structure for SAT formulas, where $λ$ is the spread parameter and $k$ is the maximum size of clauses. By the almost-independence property of containers, we prove that for formulas with $(λ,p)_k$-structures, one can distinguish between ``unsatisfiable formulas'' and ``formulas satisfying at least a $(1-δ)$-fraction of clauses'' in sub-exponential time. This shows that sufficiently spread formulas are not worst-case instances for Gap-ETH. Moreover, we show that the speedup is directly controlled by the spread parameter $λ$, yielding faster exact algorithms for SAT formulas containing a $(λ,p)_k$-structure. This result extends previous work [Zamir (STOC 2023)] to the non-uniform case.

Problem

• Spread SAT
• Boolean Satisfiability Problem
• hypergraph containers
• Gap-ETH
• clause distribution

Innovation

• hypergraph containers
• spread SAT
• weighted clause distribution
• sub-exponential algorithms
• Gap-ETH