Polynomial dependence on the doubling constant K in algorithmic PFR

Determine whether there exists an algorithm for the algorithmic Polynomial Freiman–Ruzsa tasks whose query and time complexities scale polynomially in the doubling constant K when K grows asymptotically.

Background

Throughout the paper and much of additive combinatorics, the doubling constant K is typically treated as a fixed constant since constant doubling yields strong structure. The algorithms presented have complexities optimal in n (up to logs) but do not claim polynomial dependence on K when K increases with n.

The authors explicitly highlight improving the dependence on K as an open direction, asking whether one can design algorithms whose complexities are polynomial in K for asymptotically growing K, which would strengthen the practicality of algorithmic PFR in broader regimes.