### Abstract

We provide the first hardness result for the Covering Radius Problem on lattices (CRP). Namely, we show that for any large enough p ≤ ∞ there exists a constant c_{p} > 1 such that CRP in the ℓ_{p} norm is ∏2-hard to approximate to within any constant less than c _{p}. In particular, for the case p = ∞, we obtain the constant c_{∞} = 1.5. This gets close to the constant 2 beyond which the problem is not believed to be ∏_{2}-hard. As part of our proof, we establish a stronger hardness of approximation result for the ∀∃-3-SAT problem with bounded occurrences. This hardness result might be useful elsewhere.

Original language | English (US) |
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Title of host publication | Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006 |

Pages | 145-158 |

Number of pages | 14 |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

Event | 21st Annual IEEE Conference on Computational Complexity, CCC 2006 - Prague, Czech Republic Duration: Jul 16 2006 → Jul 20 2006 |

### Other

Other | 21st Annual IEEE Conference on Computational Complexity, CCC 2006 |
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Country | Czech Republic |

City | Prague |

Period | 7/16/06 → 7/20/06 |

### Fingerprint

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

*Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006*(Vol. 2006, pp. 145-158). [1663733] https://doi.org/10.1109/CCC.2006.23

**Hardness of the covering radius problem on lattices.** / Haviv, Ishay; Regev, Oded.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006.*vol. 2006, 1663733, pp. 145-158, 21st Annual IEEE Conference on Computational Complexity, CCC 2006, Prague, Czech Republic, 7/16/06. https://doi.org/10.1109/CCC.2006.23

}

TY - GEN

T1 - Hardness of the covering radius problem on lattices

AU - Haviv, Ishay

AU - Regev, Oded

PY - 2006

Y1 - 2006

N2 - We provide the first hardness result for the Covering Radius Problem on lattices (CRP). Namely, we show that for any large enough p ≤ ∞ there exists a constant cp > 1 such that CRP in the ℓp norm is ∏2-hard to approximate to within any constant less than c p. In particular, for the case p = ∞, we obtain the constant c∞ = 1.5. This gets close to the constant 2 beyond which the problem is not believed to be ∏2-hard. As part of our proof, we establish a stronger hardness of approximation result for the ∀∃-3-SAT problem with bounded occurrences. This hardness result might be useful elsewhere.

AB - We provide the first hardness result for the Covering Radius Problem on lattices (CRP). Namely, we show that for any large enough p ≤ ∞ there exists a constant cp > 1 such that CRP in the ℓp norm is ∏2-hard to approximate to within any constant less than c p. In particular, for the case p = ∞, we obtain the constant c∞ = 1.5. This gets close to the constant 2 beyond which the problem is not believed to be ∏2-hard. As part of our proof, we establish a stronger hardness of approximation result for the ∀∃-3-SAT problem with bounded occurrences. This hardness result might be useful elsewhere.

UR - http://www.scopus.com/inward/record.url?scp=34247486199&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247486199&partnerID=8YFLogxK

U2 - 10.1109/CCC.2006.23

DO - 10.1109/CCC.2006.23

M3 - Conference contribution

AN - SCOPUS:34247486199

SN - 0769525962

SN - 9780769525969

VL - 2006

SP - 145

EP - 158

BT - Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006

ER -