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 Gr\"{o}bner 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.
翻译:根据理想的美元和多元美元,理想的会籍问题是测试是否美元。这是一个具有重要应用和臭名昭著的棘手问题。我们研究了布尔兰域受限问题产生的组合理想的“理想会籍问题”的复杂性。我们的主要结果是,我们确定了可移动性的界限。我们通过使用Gr\"{o}bner基础技术,扩大了Schaefer将布林域上所有约束性满意度问题归类为P或NP-硬体的二分点理论[STOC,1978年]。此外,我们的结果意味着有效计算Theta Body SDP放松所需的必要和充分条件,从而确定了制约语言问题的可移动性界限。本文的动机是了解最近提出的Sum-ques证据的微复杂性问题[O'Donnell, ITCS, 201717]。Raghavendra和Witz[CICM,201717]显示ICRP, 2017]显示Ideal-Is Cregresmissionality Colbility 隐含约束性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性的“会籍成员问题。