## Abstract

It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into ℓ_{1} with constant distortion. However, given an n-vertex weighted planar graph, the best upper bound on the distortion is only O(√log n), by Rao [SoCG99]. In this paper we study the case where there is a set K of terminals, and the goal is to embed only the terminals into ℓ_{1} with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into ℓ_{1}. The more general case, where the set of terminals can be covered by γ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of O(log γ) by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to O(√log γ). Since every planar graph has at most O(n) faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into ℓ_{1}. Therefore, our result provides a polynomial time O(√log γ)approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by γ faces.

Original language | English |
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Title of host publication | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 |

Editors | Shuchi Chawla |

Publisher | Association for Computing Machinery |

Pages | 1945-1954 |

Number of pages | 10 |

ISBN (Electronic) | 9781611975994 |

State | Published - 2020 |

Externally published | Yes |

Event | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States Duration: 5 Jan 2020 → 8 Jan 2020 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2020-January |

### Conference

Conference | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 |
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Country/Territory | United States |

City | Salt Lake City |

Period | 5/01/20 → 8/01/20 |

### Bibliographical note

Publisher Copyright:Copyright © 2020 by SIAM

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