Ideal Membership Problem for Boolean Minority and Dual Discriminator

TL;DR

This paper proves that the polynomial Ideal Membership Problem for Boolean CSPs with minority and dual discriminator polymorphisms can be solved in polynomial time, completing the classification of tractability for these problems.

Contribution

It establishes polynomial-time algorithms for IMP$_d$ in Boolean CSPs with minority and dual discriminator polymorphisms, filling previous gaps in complexity classification.

Findings

IMP$_d$ for Boolean minority polymorphism is polynomial-time solvable.

IMP$_d$ for dual discriminator polymorphism over any finite domain is polynomial-time solvable.

Results have applications in Nullstellensatz and Sum-of-Squares proof systems.

Abstract

We consider the polynomial Ideal Membership Problem (IMP) for ideals encoding combinatorial problems that are instances of CSPs over a finite language. In this paper, the input polynomial $f$ has degree at most $d=O(1)$ (we call this problem IMP$_d$). We bridge the gap in \cite{MonaldoMastrolilli2019} by proving that the IMP$_d$ 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 IMP$_d$. We also prove that the proof of membership for the IMP$_d$ 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.