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arXiv · 2507.17878

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

Benjamin Bedert·Tamio-Vesa Nakajima·Karolina Okrasa·Stanislav Živný
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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.

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Benjamin Bedert, Tamio-Vesa Nakajima, Karolina Okrasa, Stanislav Živný. 2025-07-23. Strong Sparsification for 1-in-3-SAT via Polynomial Freiman-Ruzsa. https://arxiv.org/abs/2507.17878

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