Strong Sparsification for 1-in-3-SAT via Polynomial Freiman-Ruzsa
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 . 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.
Keywords
Cite
@article{arxiv.2507.17878,
title = {Strong Sparsification for 1-in-3-SAT via Polynomial Freiman-Ruzsa},
author = {Benjamin Bedert and Tamio-Vesa Nakajima and Karolina Okrasa and Stanislav Živný},
journal= {arXiv preprint arXiv:2507.17878},
year = {2026}
}
Comments
Full version of a FOCS'25 paper; v2 has more results; v3 proves Conjecture 33 from v2, giving a tight bound on strong sparsifiability of 1-in-k-SAT (in possibly exponential time); v4 improves presentation and makes constants more explicit