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

We introduce a new notion of sparsification, called 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 F 𝑑 2 . This result, obtained using the recent 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 Živný, APPROX 2024). We also investigate the existence of strong sparsification algorithms for other constraint satisfaction problems.