## Abstract

Suppose that every k points in a n point metric space X are D-distortion embeddable into ℓ1. We give upper and lower bounds on the distortion required to embed the entire space X into ℓ1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into ℓ1 with distortion O(D X log(n/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For D = 1 + δ we give a lower bound of ω(log(n/k)/log(1/δ)); and for D =1, we give a lower bound of ω(logn/(log k + loglogn)). Our bounds significantly improve on the results of Arora et al. who initiated a study of these questions.

Original language | English (US) |
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Pages (from-to) | 2487-2512 |

Number of pages | 26 |

Journal | SIAM Journal on Computing |

Volume | 39 |

Issue number | 6 |

DOIs | |

State | Published - May 19 2010 |

## Keywords

- Global local properties
- Lift and project
- Metric embeddings

## ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)