The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if f ϵ I . This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the IMP for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer’s dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems (CSPs) over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body Semi-Definite Programming (SDP) relaxations, identifying therefore the borderline of tractability for constraint language problems. This article is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares (SoS) proofs [O’Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the IMP tractability for combinatorial ideals implies bounded coefficients in SoS proofs.

Local zero estimates and effective division in rings of algebraic power series

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

Gröbner–Shirshov Bases for Replicated Algebras

We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).