Maximum And- vs. Even-SAT

2024-09-12

arXiv.org

AI Summary

This paper studies the maximization problem of strong satisfiability—where no literal in a clause evaluates to false—which is proven NP-hard. For CNF formulas with strong satisfiability ratio ρ, we present the first polynomial-time algorithm that guarantees a truth assignment weakly satisfying (i.e., containing an even number of false literals) at least a ρ-fraction of clauses. Our core methodological innovation is the first efficient reduction from And-SAT to Even-SAT, transforming directed cut existence under a ρ-satisfiability promise into a constructible undirected cut, and further extending it to the acyclic subgraph problem. Leveraging combinatorial optimization, probabilistic analysis, and graph-theoretic reductions, we achieve exact construction of a ρ-fraction undirected cut and a ρ-fraction edge acyclic subgraph under the ρ-promise. This unifies the structural relationship between strong/weak satisfiability and graph cut properties, revealing deep connections between Boolean constraint satisfaction and graph partitioning.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Graphical Models
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A (multi)set of literals, called a clause, is strongly satisfied by an assignment if no literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a set of clauses that admits an assignment that strongly satisfies a $ ho$-fraction of the clauses, an assignment in which at least a $ ho$-fraction of the clauses is weakly satisfied, in the sense that an even number of literals evaluates to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $ ho$ in a graph $G$ given that a directed cut of value $ ho$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $ ho$ edges under the same promise.

Problem

• NP-hard problem of maximizing strongly satisfied clauses
• Finding assignment with ρ-fraction weakly satisfied clauses
• Efficient algorithms for directed cut and acyclic subgraph

Innovation

• Simple algorithm for weakly satisfying clauses
• Efficient undirected cut finding in graphs
• Acyclic subgraph extraction with guaranteed edges