## Abstract

We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into L1. In particular, we show that for every n ≥ 1, there is an n-point metric space of negative type that requires a distortion of ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of (log log n) ^{1/6-o}(1) due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into l _{1}, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 53-62]. We also show that embedding the edit distance metric on {0, 1} ^{n} into L _{1} requires a distortion of Ω(log n). This result improves a very recent (log n) ^{1/2-o}(1) lower bound by Khot and Naor [Nonembeddability theorems via Fourier analysis, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 101-112].

Original language | American English |
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Pages (from-to) | 2487-2498 |

Number of pages | 12 |

Journal | SIAM Journal on Computing |

Volume | 38 |

Issue number | 6 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

## Keywords

- Approximation algorithms
- Edit distance
- Graph partitioning
- Integrality gap
- Metric embeddings
- Negative type metrics
- Semidefinite programming relaxation

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