On the complexity of CSP-based ideal membership problems

DOI10.1145/3519935.3520063arXiv2011.03700OpenAlexW3099168587WikidataQ130978292 ScholiaQ130978292MaRDI QIDQ6083496

Andrei Bulatov, Akbar Rafiey

Publication date: 8 December 2023

Published in: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (Search for Journal in Brave)

Abstract: In this paper we consider the Ideal Membership Problem (IMP for short), in which we are given real polynomials $f_0,f_1,dots,f_k$ and the question is to decide whether $f_0$ belongs to the ideal generated by $f_1,dots,f_k$. 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 $CSP(Gamma)$, that is, CSPs in which the type of constraints is limited to relations from a set $Gamma$. 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 in terms of bit complexity of sum-of-squares (SOS) proofs and their automatizability, and studying (construction of) theta bodies of combinatorial problems.

Full work available at URL: https://arxiv.org/abs/2011.03700

zbMATH Keywords

Groebner basisconstraint satisfaction problemspolymorphismspolynomial ideal membership

Mathematics Subject Classification ID

Theory of computing (68Qxx)

Related Items (4)

Unnamed Item ⋮ Unnamed Item ⋮ Short proofs of ideal membership ⋮ Bi-arc digraphs: recognition algorithm and applications

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