Strong Sparsification for 1-in-3-SAT via Polynomial Freiman-Ruzsa
Benjamin Bedert, Tamio-Vesa Nakajima, Karolina Okrasa, Stanislav Živný
TL;DR
This work introduces strong sparsification, merging variables to preserve solution structure in CSPs, and proves a polynomial-time algorithm for monotone $1$-in-$3$-SAT that achieves a subquadratic reduction to $O(n^{2-\varepsilon})$ constraints, with $\varepsilon\approx 0.0028$. The core technical advance leverages additive combinatorics—combining Balog–Szemerédi–Gowers and Polynomial Freiman–Ruzsa—to bound the aggregate merge potential in vector spaces over $\mathbb{F}_2^d$, including a special-case polynomial-method bound. As an application, the authors improve the state-of-the-art for linearly-ordered colourings of 3-uniform hypergraphs, obtaining an LO $0.999\log_2 n$-colouring from LO 2-colourability. The paper also develops a framework connecting monotone to non-monotone sparsification and provides general lower bounds, showing that strong sparsification cannot be achieved for many CSPs beyond certain exponents, thereby clarifying the limits of this approach in CSP sparsification.
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).