### 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) |
---|---|

Pages (from-to) | 843-873 |

Number of pages | 31 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### Keywords

- Graph coloring
- Hardness of approximation
- Unique games

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science(all)

### Cite this

*SIAM Journal on Computing*,

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

**Conditional hardness for approximate coloring.** / Dinur, Irit; Mossel, Elchanan; Regev, Oded.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 39, no. 3, pp. 843-873. https://doi.org/10.1137/07068062X

}

TY - JOUR

T1 - Conditional hardness for approximate coloring

AU - Dinur, Irit

AU - Mossel, Elchanan

AU - Regev, Oded

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Graph coloring

KW - Hardness of approximation

KW - Unique games

UR - http://www.scopus.com/inward/record.url?scp=69049114323&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69049114323&partnerID=8YFLogxK

U2 - 10.1137/07068062X

DO - 10.1137/07068062X

M3 - Article

VL - 39

SP - 843

EP - 873

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 3

ER -