On the complexity of CSP-based ideal membership problems

In this paper we consider the Ideal Membership Problem (IMP for short), in which we are given real polynomials f 0 , f 1 , . . . , f k and the question is to decide whether f 0 belongs to the ideal generated by f 1 , . . . , f k . In the more stringent version the task is also to find a proof of this fact. The IMP underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. In the majority of such applications the IMP involves so called combinatorial ideals that arise from a variety of discrete combinatorial problems. This restriction makes the IMP significantly easier and in some cases allows for an efficient algorithm to solve it. The first part of this paper follows the work of Mastrolilli [SODA'19] who initiated a systematic study of IMPs arising from Constraint Satisfaction Problems (CSP) of the form CSP(Γ), that is, CSPs in which the type of constraints is limited to relations from a set Γ. We show that many CSP techniques can be translated to IMPs thus allowing us to significantly improve the methods of studying the complexity of the IMP. We also develop universal algebraic techniques for the IMP that have been so useful in the study of the CSP. This allows us to prove a general necessary condition for the tractability of the IMP, and three sufficient ones. The sufficient conditions include IMPs arising from systems of linear equations over GF(p), p prime, and also some conditions defined through special kinds of polymorphisms. Our work has several consequences and applications. First, we introduce a variation of the IMP and based on this propose a unified framework, different from the celebrated Buchberger's algorithm, to construct a bounded degree Gröbner Basis. Our algorithm, combined with the universal algebraic techniques, leads to polynomial-time construction of Gröbner Basis for many combinatorial problems. Second, we also study a recently posed question by O'Donnell [ITCS'17] on the bit complexity of sum-of-squares (SOS) proofs and their automatizability. We prove that for almost all the CSP-based ideals SOS proofs are automatizable which improves upon the result of Raghavendra and Weitz [ICALP'17]. Furthermore, in the case where the IMP part is well behaved, we propose automatizable SOS proofs of nonnegativity with much relaxed degree restrictions. Finally, we provide a unifying framework that studies (construction of) theta bodies of combinatorial problems through the universal algebra of the constraint languages and present several positive results for problems such as Stable Set, Binary Matroids, H-Coloring, Min/Max Ones, and Strict CSPs.