## Abstract

Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a'small-complexity' graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics: 1)Construction of a light subset spanner. Given a subset of vertices called terminals, and epsilon, in polynomial time we construct a sub graph that preserves all pairwise distances between terminals up to a multiplicative 1+ epsilon factor, of total weight at most O {epsilon}(1) times the weight of the minimal Steiner tree spanning the terminals. 2)Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion epsilon D. Namely, given a minor-free graph G= (V, E, w) of diameter D, and parameter epsilon, we construct a distribution mathcal{D} over dominating metric embeddings into treewidth-O {epsilon}(log n) graphs such that forall u, v in V, mathbb{E} {f sim mathcal{D}}[d {H}(f(u), f(v))] leq d {G}(u, v)+ epsilon D. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for bounded-capacity vehicle routing in minor-free metrics; (3) the first efficient approximation scheme for bounded-capacity vehicle routing on bounded genus metrics. En route to the latter result, we design the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth).

Original language | English |
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Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |

Publisher | IEEE Computer Society |

Pages | 589-600 |

Number of pages | 12 |

ISBN (Electronic) | 9781728196213 |

DOIs | |

State | Published - Nov 2020 |

Externally published | Yes |

Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2020-November |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
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Country/Territory | United States |

City | Virtual, Durham |

Period | 16/11/20 → 19/11/20 |

### Bibliographical note

Publisher Copyright:© 2020 IEEE.

## Keywords

- metric embedding
- minor-free graphs
- spanners
- travelling salesperson problem
- vehicle routing