We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a combination of the Gaussian elimination technique (i.e., solving a system of linear equations over an Abelian group) and the semidefinite programming relaxation. We complement our algorithm with a matching dictator vs. quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel invariance principle, which we call the mixed invariance principle. Our mixed invariance principle is an extension of the invariance principle of Mossel, O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial role in Raghavendra's work. The mixed invariance principle allows one to relate 3-wise correlations over discrete probability spaces with expectations over spaces that are a mixture of Guassian spaces and Abelian groups, and may be of independent interest.

翻译：我们为所有最大约束满足问题提出了一种算法与难解性分析框架，并针对一大类谓词展示了该框架的应用。此框架拓展了Raghavendra [STOC, 2008]的工作——其研究主要针对近似可满足的最大约束满足问题。我们的框架基于一种新型混合近似算法，该算法融合了高斯消元技术（即在阿贝尔群上求解线性方程组）与半定规划松弛方法。我们通过设计一个具有完全完备性的匹配型"独裁函数与拟随机测试"来完善算法体系。对该测试的分析基于一种新颖的不变性原理，我们称之为混合不变性原理。此原理是对Mossel、O'Donnell与Oleszkiewicz [Annals of Mathematics, 2010]提出的不变性原理的扩展——该原理在Raghavendra的研究中具有关键作用。混合不变性原理能够将离散概率空间上的三阶相关性，与高斯空间和阿贝尔群混合空间上的数学期望建立联系，这一原理本身可能具有独立的研究价值。