On Approximability of Satisfiable k-CSPs: V
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arXiv
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| Format: | Preprint |
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2024
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| author | Bhangale, Amey Khot, Subhash Minzer, Dor |
| author_facet | Bhangale, Amey Khot, Subhash Minzer, Dor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Approximability of Satisfiable k-CSPs: V Bhangale, Amey Khot, Subhash Minzer, Dor Computational Complexity 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. |
| title | On Approximability of Satisfiable k-CSPs: V |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2408.15377 |