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). We also investigate the existence of strong sparsification algorithms for other constraint satisfaction problems.

翻译：我们引入了稀疏化的一种新概念，称为“强稀疏化”，在此过程中约束不会被移除，但变量可以被合并。作为我们的主要结果，我们提出了针对1-in-3-SAT的一种强稀疏化算法。该算法的正确性依赖于对$\mathbb{F}_2^d$中某些向量集合大小的亚二次上界的确立。这一结果得益于近期提出的《多项式Freiman-Ruzsa定理》(Gowers, Green, Manners 与 Tao, Ann. Math. 2025)，可能具有独立的研究价值。作为应用，我们改进了针对3-一致超图线性序着色近似的最新算法(Håstad, Martinsson, Nakajima 与 Živný, APPROX 2024)。我们还探讨了其他约束满足问题中强稀疏化算法的存在性。