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：我们提出了一个面向所有Max-CSP问题的算法与下界框架，并在一大类谓词上进行了验证。该框架扩展了Raghavendra [STOC, 2008] 针对几乎可满足Max-CSP问题的工作。我们的框架基于一种新的混合近似算法，该算法结合了高斯消元技术（即在阿贝尔群上求解线性方程组）与半定规划松弛。我们通过一个具有完美完备性的匹配的独裁者 vs. 拟随机测试来补充该算法。该测试的分析基于一个新的不变性原理，我们称之为混合不变性原理。该原理是Mossel、O'Donnell和Oleszkiewicz [Annals of Mathematics, 2010] 提出的不变性原理的扩展——后者在Raghavendra的工作中发挥了关键作用。混合不变性原理能够将离散概率空间上的3元相关性，与高斯空间和阿贝尔群混合空间上的期望联系起来，且可能具有独立的研究价值。