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 ${\mathbb{F}}_2^d$. 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.

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