Strong Sparsification for 1-in-3-SAT via Polynomial Freiman-Ruzsa

2025-07-23

AI Summary

This paper addresses the 1-in-3-SAT problem by introducing a novel “strong sparsification” paradigm: instead of deleting constraints, it compresses instances via variable merging. The method leverages additive combinatorial structure analysis of vector sets over the finite field 𝔽₂ᵈ and, for the first time, incorporates the polynomial Freiman–Ruzsa theorem into sparsification frameworks—yielding a subquadratic upper bound on set size that ensures algorithmic correctness. Based on this, we design the first strong sparsification algorithm that preserves all constraints while significantly reducing the number of variables. Furthermore, we apply this algorithm to improve the approximation ratio for linear ordering coloring of 3-uniform hypergraphs, surpassing the best-known prior results. The core contributions are (i) establishing a theoretical foundation for variable-merging-based sparsification, and (ii) providing new combinatorial tools and an algorithmic paradigm bridging constraint satisfaction problems and hypergraph coloring.

Technology Category

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

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce a new notion of sparsification, called emph{strong sparsification}, in which constraints are not removed but variables can be merged. As our main result, we present a strong sparsification algorithm for 1-in-3-SAT. The correctness of the algorithm relies on establishing a sub-quadratic bound on the size of certain sets of vectors in $mathbb{F}_2^d$. This result, obtained using the recent emph{Polynomial Freiman-Ruzsa Theorem} (Gowers, Green, Manners and Tao, Ann. Math. 2025), could be of independent interest. As an application, we improve the state-of-the-art algorithm for approximating linearly-ordered colourings of 3-uniform hypergraphs (Håstad, Martinsson, Nakajima and{Ž}ivn{ý}, APPROX 2024).

Problem

• Develops strong sparsification for 1-in-3-SAT by merging variables
• Proves sub-quadratic vector set bounds using Polynomial Freiman-Ruzsa Theorem
• Improves approximation algorithms for 3-uniform hypergraph colorings

Innovation

• Introduces strong sparsification via variable merging
• Uses Polynomial Freiman-Ruzsa Theorem for bounds
• Improves hypergraph coloring approximation algorithms