### Abstract

We study the AprxColoring(q, Q) problem: Given a graph G, decide whether x(G)≤ q or x(G) ≥ Q. We present hardness results for this problem for any constants 3 ≤ q <Q. For q ≥4, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775]. For q = 3, we base our hardness result on a certain "⋉-shaped" variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz ["Noise stability of functions with low influences: Invariance and optimality, " Ann. of Math. (2), to appear] and should have wider applicability.

Original language | English (US) |
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Pages (from-to) | 843-873 |

Number of pages | 31 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - Aug 27 2009 |

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### Keywords

- Graph coloring
- Hardness of approximation
- Unique games

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

### Cite this

*SIAM Journal on Computing*,

*39*(3), 843-873. https://doi.org/10.1137/07068062X