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

TL;DR

This paper introduces a novel strong sparsification method for 1-in-3-SAT, leveraging the Polynomial Freiman-Ruzsa Theorem to improve algorithms for hypergraph coloring and explore broader CSP applications.

Contribution

It presents the first strong sparsification algorithm for 1-in-3-SAT using advanced additive combinatorics techniques, with implications for hypergraph coloring approximations.

Findings

Established a sub-quadratic bound on vector sets in $ ext{F}_2^d$

Improved approximation algorithms for linearly-ordered hypergraph colorings

Demonstrated potential for strong sparsification in other CSPs

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 (H{\aa}stad, Martinsson, Nakajima and{\v{Z}}ivn{\'{y}}, APPROX 2024). We also investigate the existence of strong sparsification algorithms for other constraint satisfaction problems.