The Ideal Membership Problem (IMP) tests if an input polynomial $f\in \mathbb{F}[x_1,\dots,x_n]$ with coefficients from a field $\mathbb{F}$ belongs to a given ideal $I \subseteq \mathbb{F}[x_1,\dots,x_n]$. 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 ``hard'' (NP-hard) and ``easy'' (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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布尔少数派的理想成员问题

理想成员问题 (IMP) 测试输入多项式 $f\in \mathbb{F}[x_1,\dots,x_n]$ 是否属于给定的理想 $I \ subseteq \mathbb{F}[x_1,\dots,x_n]$。这是一个众所周知的基本问题，具有许多重要的应用程序，尽管在一般情况下非常棘手。在本文中，我们考虑了多项式理想编码组合问题的 IMP，其中输入多项式 $f$ 的度数最多为 $d=O(1)$（我们称这个问题为 IMP$_d$）。最近在有限域上的约束满足问题 [Bulatov FOCS'17, Zhuk FOCS'17] 中实现了“hard”（NP-hard）和“easy”（多项式时间）IMP 之间的二分法结果（这等效于到 IMP$_0$) 和 IMP$_d$ 用于布尔域 [Mastrolilli SODA'19]，两者都基于通过称为多态性的函数对 IMP 进行分类。五个多态性的 IMP$_d$ 的复杂性已在 [Mastrolilli SODA'19] 中解决，而对于三元少数多态性，它被错误地声明为已被先前的结果解决。事实上，三元少数多态性的 IMP$_d$ 的复杂性是开放的。在本文中，我们通过证明约束在少数多态性下闭合的布尔组合理想的 IMP$_d$ 可以在多项式时间内求解来提供缺失的链接。该结果与 [Mastrolilli SODA'19] 中的结果一起，完成了对 IMP$_d$ 的精确易处理性边界的识别，用于布尔域上的约束问题。本文旨在了解 O'Donnell [ITCS'17] 提出的平方和证明的位复杂性问题。Raghavendra 和 Weitz [ICALP'17] 展示了组合理想的 IMP$_d$ 易处理性如何暗示平方和证明中的有界系数。