## Abstract

We develop a polynomial time ω(1/R log R) approximate algorithm for Max 2CSP-.R, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size R, and we want to find an assignment to the variables that maximizes the number of satisfied constraints. Assuming the Unique Games Conjecture, this is the best possible approximation up to constant factors. Previously, a 1/R-approximate algorithm was known, based on linear programming. Our algorithm is based on semidefinite programming. The Semidefinite Program that we use has an almost-matching integrality gap. For the more general Max kCSP-R, in which each constraint involves k variables, each ranging over a set of size R, it was known that the best possible approximation is of the order of k/R^{k-1}, provided that k is sufficiently large compared to R; our algorithm shows that the bound k/R^{k-1} is not tight for k = 2.

Original language | American English |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 1705-1714 |

Number of pages | 10 |

ISBN (Electronic) | 9781510819672 |

State | Published - 2016 |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: 10 Jan 2016 → 12 Jan 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 3 |

### Conference

Conference | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country/Territory | United States |

City | Arlington |

Period | 10/01/16 → 12/01/16 |

### Bibliographical note

Funding Information:Part of this work was done while visiting the Simons Institute. Supported by Israeli Science Fund Grant No. 1692/13, and Binational Science Foundation Grant No. 2012220. Part of this work was done while visiting the Simons Institute. This material is based upon work supported by the National Science Foundation under Grant No. 1423452. This material is based upon work supported by the National Science Foundation under Grant No. 1216642 and by the US-Israel Binational Science Foundation under Grant No. 2010451. blankline%

Publisher Copyright:

© (2016) by SIAM: Society for Industrial and Applied Mathematics.