Under what condition is a random constraint satisfaction problem hard to refute by the sum-of-squares (SoS) algorithm? A sufficient condition is t-wise uniformity, that is, each constraint has a t-wise uniform distribution of satisfying assignments, as shown by the lower bounds of Kothari, Mori, O'Donnell, and Witmer (STOC 2017). This condition is also necessary for random CSPs given by a predicate and uniformly random literals, due to the constant-degree SoS refutation of Allen, O'Donnell, and Witmer (FOCS 2015). For higher degree, Raghavendra, Rao, and Schramm (STOC 2017) gave a refutation for Boolean random CSPs with uniformly random literals, matching the lower bounds optimally in terms of the three-way tradeoff between constraint density, SoS degree, and strength of refutation. Two long-standing open problems are to find a more general sufficient condition for SoS lower bounds, and to refute similar random CSPs not involving literals. We show that for a general random k-CSP, the necessary and sufficient hardness condition is not t-wise uniformity, but t-wise independence. We generalize the optimal three-way tradeoff to any random k-CSP, without assuming a Boolean domain or uniformly random literals. Our analysis involves new Kikuchi matrices for odd order and for asymmetric tensors. It also uses the global correlation rounding technique of Barak, Raghavendra, and Steurer (FOCS 2011). To avoid the running-time penalty of this technique, we also give a spectral refutation algorithm.

翻译：对于随机约束满足问题，在何种条件下，平方和（SoS）算法难以对其进行反驳？Kothari、Mori、O'Donnell 与 Witmer（STOC 2017）的下界表明，一个充分条件是t-阶一致性，即每个约束的满足分配具有t-阶一致分布。对于由谓词和均匀随机文字给出的随机CSP，该条件也是必要的，这得益于 Allen、O'Donnell 与 Witmer（FOCS 2015）的常数度SoS反驳。对于更高次数，Raghavendra、Rao 与 Schramm（STOC 2017）给出了带均匀随机文字的布尔随机CSP的反驳，在约束密度、SoS次数与反驳强度三者权衡的最优性上匹配了下界。两个长期未解决的开放问题是：寻找更一般的充分条件以支持SoS下界，以及反驳不涉及文字的类似随机CSP。我们证明，对于一般的随机k-CSP，必要且充分的困难条件并非t-阶一致性，而是t-阶独立性。我们将最优的三方权衡推广到任意随机k-CSP，无需假设布尔论域或均匀随机文字。我们的分析涉及新的奇数阶及非对称张量的Kikuchi矩阵，并采用了Barak、Raghavendra与Steurer（FOCS 2011）的全局相关舍入技术。为避免该技术带来的运行时间惩罚，我们还给出了一种谱反驳算法。