Ideal Membership Problem for Boolean Minority and Dual Discriminator

The polynomial Ideal Membership Problem (IMP) tests if an input polynomial f ∈ F[x1, . . . , xn] with coefficients from a field F belongs to a given ideal I ⊆ F[x1, . . . , xn]. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial f has degree at most d = O(1) (we call this problem IMPd). A dichotomy result between “hard” (NP-hard) and “easy” (polynomial time) IMPs was achieved for Constraint Satisfaction Problems over finite domains [6, 34] (this is equivalent to IMP0) and IMPd for the Boolean domain [23], both based on the classification of the IMP through functions called polymorphisms. For the latter result, there are only six polymorphisms to be studied in order to achieve a full dichotomy result for the IMPd. The complexity of the IMPd for five of these polymorphisms has been solved in [23] whereas for the ternary minority polymorphism it was incorrectly declared in [23] to have been resolved by a previous result. In this paper we provide the missing link 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 precise borderline of tractability for the IMPd for constrained problems over the Boolean domain. We also prove that the proof of membership for the IMPd for problems constrained by the dual discriminator polymorphism over any finite domain can also be found in polynomial time. Bulatov and Rafiey [8] recently proved that the IMPd for this polymorphism is decidable in polynomial time, without needing a proof of membership. Our result gives a proof of membership and can be used in applications such as Nullstellensatz and Sum-of-Squares proofs. 2012 ACM Subject Classification Mathematics of computing → Gröbner bases and other special bases; Mathematics of computing → Combinatoric problems

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