## Abstract

We study the problem of embedding weighted graphs of pathwidth k into ℓ_{p} spaces. Our main result is an O(k^{min} {1/p^{, 1}/2 ^{}})-distortion embedding. For p = 1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p > 1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p > 2 our embedding also implies improved distortion on bounded treewidth graphs (O((k log n)^{1}/^{p})). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.

Original language | English |
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Title of host publication | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Monika Henzinger, David Kempe, Ilias Diakonikolas |

Publisher | Association for Computing Machinery |

Pages | 912-919 |

Number of pages | 8 |

ISBN (Electronic) | 9781450355599 |

DOIs | |

State | Published - 20 Jun 2018 |

Externally published | Yes |

Event | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
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Country/Territory | United States |

City | Los Angeles |

Period | 25/06/18 → 29/06/18 |

### Bibliographical note

Funding Information:Arnold Filtser is Partially supported by the Lynn and William Frankel Center for Computer Sciences, ISF grant 1817/17, and by BSF Grant 2015813. Anupam Gupta is supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790. Ofer Neiman is supported in part by ISF grant 1817/17, and by BSF Grant 2015813.

Publisher Copyright:

© 2018 Association for Computing Machinery.

## Keywords

- Metric embeddings
- Normed spaces
- Pathwidth
- Shortest path decomposition
- Treewidth