## Abstract

Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minor-free graphs of diameter D into graphs of treewidth O?(log n) with expected additive distortion +?D. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to O?(log log n)2. As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from nO?(log (n)) to nO?(log log (n))3. Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter D into graphs of treewidth O((log log n)2)/?) and additive distortion +?D that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time. A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth O(log log n) and hop-diameter O(log log n). This result may be of independent interest.

Original language | English |
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Title of host publication | Proceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022 |

Publisher | IEEE Computer Society |

Pages | 1081-1092 |

Number of pages | 12 |

ISBN (Electronic) | 9781665455190 |

DOIs | |

State | Published - 2022 |

Event | 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 - Denver, United States Duration: 31 Oct 2022 → 3 Nov 2022 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2022-October |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 |
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Country/Territory | United States |

City | Denver |

Period | 31/10/22 → 3/11/22 |

### Bibliographical note

Publisher Copyright:© 2022 IEEE.

## Keywords

- PTAS
- metric embedding
- minor-free graphs
- planar graphs
- vehicle routing problem