IDEAL MEMBERSHIP PROBLEM FOR BOOLEAN MINORITY AND DUAL DISCRIMINATOR

Journal article, 2025

We consider the polynomial ideal membership problem (IMP) for ideals encoding combinatorial problems that are instances of constraint satisfaction problems over a finite language. In this paper, the input polynomial f has degree at most d = O(1) (we call this problem IMPd). We bridge the gap in [M. Mastrolilli, The complexity of the ideal membership problem for constrained problems over the Boolean domain, in SODA '19, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, PA, Society for Industrial and Applied Mathematics, 2019, pp. 456-475] by proving that the IMPd for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the tractability for the Boolean IMPd. We also prove that the proof of membership for the IMPd for problems constrained by the dual discriminator polymorphism over any finite domain can be found in polynomial time. Our results can be used in applications such as Nullstellensatz and sum-of-squares proofs.