Ideal Membership Problem for Boolean Minority
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The Ideal Membership Problem (IMP) tests if an input polynomial fβπ½[x_1,β¦,x_n] with coefficients from a field π½ belongs to a given ideal I βπ½[x_1,β¦,x_n]. 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 IMP_d). A dichotomy result between βhardβ (NP-hard) and βeasyβ (polynomial time) IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Bulatov FOCS'17, Zhuk FOCS'17] (this is equivalent to IMP_0) and IMP_d for the Boolean domain [Mastrolilli SODA'19], both based on the classification of the IMP through functions called polymorphisms. The complexity of the IMP_d for five polymorphisms has been solved in [Mastrolilli SODA'19] whereas for the ternary minority polymorphism it was incorrectly declared to have been resolved by a previous result. As a matter of fact the complexity of the IMP_d for the ternary minority polymorphism is open. In this paper we provide the missing link 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 result, along with the results in [Mastrolilli SODA'19], completes the identification of the precise borderline of tractability for the IMP_d for constrained problems over the Boolean domain. This paper is motivated by the pursuit of understanding the issue of bit complexity of Sum-of-Squares proofs raised by O'Donnell [ITCS'17]. Raghavendra and Weitz [ICALP'17] show how the IMP_d tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.
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