### Abstract

This paper studies how well the standard LP relaxation approximates a k-ary constraint satisfaction problem (CSP) on label set [L]. We show that, assuming the Unique Games Conjecture, it achieves an approximation within O(k^{3} ・ log L) of the optimal approximation factor. In particular we prove the following hardness result: let I be a k-ary CSP on label set [L] with constraints from a constraint class C, such that it is a (c, s)-integrality gap for the standard LP relaxation. Then, given an instance H with constraints from C, it is NP-hard to decide whether, (Formula Presented.) assuming the Unique Games Conjecture. We also show the existence of an efficient LP rounding algorithm Round such that given an instance H from a permutation invariant constraint class C which is a (c, s)-rounding gap for Round, it is NP-hard to decide whether, (Formula Presented.) assuming the Unique Games Conjecture.

Original language | English (US) |
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Title of host publication | Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings |

Editors | Magnus M. Halldorsson, Naoki Kobayashi, Bettina Speckmann, Kazuo Iwama |

Publisher | Springer Verlag |

Pages | 822-833 |

Number of pages | 12 |

ISBN (Print) | 9783662476710 |

DOIs | |

State | Published - 2015 |

Event | 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 - Kyoto, Japan Duration: Jul 6 2015 → Jul 10 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9134 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 |
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Country | Japan |

City | Kyoto |

Period | 7/6/15 → 7/10/15 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings*(pp. 822-833). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9134). Springer Verlag. https://doi.org/10.1007/978-3-662-47672-7_67