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).

翻译：本文引入了一种称为\emph{强稀疏化}的新稀疏化概念，其中约束条件不会被移除，但变量可以被合并。作为主要结果，我们提出了一种针对1-in-3-SAT问题的强稀疏化算法。该算法的正确性依赖于在$\mathbb{F}_2^d$中特定向量集合的大小上建立一个次二次上界。这一结果通过运用近期提出的\emph{多项式Freiman-Ruzsa定理}（Gowers、Green、Manners与Tao，Ann. Math. 2025）获得，可能具有独立的研究价值。作为应用，我们改进了近似3-一致超图线性有序着色问题的最先进算法（Håstad、Martinsson、Nakajima与{Ž}ivn{ý}，APPROX 2024）。