On Approximability of Satisfiable k-CSPs: V
<|reference_start|>On Approximability of Satisfiable k-CSPs: V: 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.<|reference_end|>
@article{bhangale2024on,
    title={On Approximability of Satisfiable k-CSPs: V},
    author={Amey Bhangale, Subhash Khot and Dor Minzer},
    journal={arXiv preprint arXiv:2408.15377},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.15377},
    primaryClass={cs.CC}
}