布尔域上约束问题理想隶属度问题的复杂性
The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain
摘要：
给定一个理想的$I$和一个多项式$f$，理想成员问题是测试是否$f\in I$。这个问题是一个基本的算法问题，具有重要的应用和众所周知的棘手。我们研究的复杂性的理想成员资格问题的组合理想，所产生的约束问题的布尔域。作为我们的主要结果，我们确定了可处理性的边界。通过使用Gröbner基技术，我们扩展了Schaefer的二分法定理[STOC，1978]，该定理将布尔域上的所有约束满足问题分类为P或NP-困难。此外，我们的结果意味着充要条件的有效计算Theta体SDP松弛，因此确定边界的易处理性约束语言问题。本文的动机是追求理解最近提出的平方和证明的位复杂性问题[O 'Donnell，ITCS，2017]。Raghavendra和Weitz [ICALP，2017]展示了组合理想的理想成员问题易处理性如何在平方和证明中暗示有界系数。
英文摘要：
Given an ideal $I$ and a polynomial $f$ the Ideal Membership Problem is to test if $f\in I$. This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the Ideal Membership Problem 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 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 SDP relaxations, identifying therefore the borderline of tractability for constraint language problems. This paper is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares proofs [O'Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the Ideal Membership Problem tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.
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