## Abstract

We consider the problem of embedding a finite set of points {x1, . . . , xn} 2 Rd that satisfy 22 triangle inequalities into 1, when the points are approximately low-dimensional. Goemans (unpublished, appears in [20]) showed that such points residing in exactly d dimensions can be embedded into 1 with distortion at most p d. We prove the following robust analogue of this statement: if there exists a r-dimensional subspacesuch that the projections onto this subspace satisfy P i,j2[n] kxi - xjk 22 (1) P i,j2[n] kxi - xjk 22 , then there is an embedding of the points into 1 with O(p r) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(p r) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies r(G)/n(1)SDP (G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [7, 6].

Original language | American English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017 |

Editors | Jose D. P. Rolim, Klaus Jansen, David P. Williamson, Santosh S. Vempala |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770446 |

DOIs | |

State | Published - 1 Aug 2017 |

Event | 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, United States Duration: 16 Aug 2017 → 18 Aug 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 81 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 |
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Country/Territory | United States |

City | Berkeley |

Period | 16/08/17 → 18/08/17 |

### Bibliographical note

Publisher Copyright:© Yuval Rabani and Rakesh Venkat.

## Keywords

- Approximation Algorithms
- Metric Embeddings
- Negative type metrics
- Sparsest Cut