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.
翻译:在本文中,我们考虑Ideal Assist Reform (IMP for short) 问题。 IMP是许多基于多边协议的验证系统的基础, 比如 Nullstellensatz, olynamial Calulus, 和 Sum- quares 。 在大多数这样的应用中, IMP 涉及由离散的组合问题产生的所谓的组合式理想。 这一限制使得IMP大大容易, 在某些情况下, 也能够找到一个有效的解答。 在更严格的版本中, 任务也是找到一个事实的证明。 IMP( SO'19) 的工作是启动对IMPs 的系统( IMP) 的系统性研究, IMP 和 Sum- squal- squal- squal) 。 在这种应用中, IMP 也能够让 C- Proferal 出现一定的硬性要求。 IMP 的特性可以让 Cal 的系统通过 Serveal 系统( Serviews) 和 Ex ex ex ex ex real ex ex ex ex ex) a exmess the C.