On light spanners, low-treewidth embeddings and efficient traversing in minor-free graphs

Vincent Cohen-Addad, Arnold Filtser, Philip N. Klein, Hung Le

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

18 Scopus citations

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 languageEnglish
Title of host publicationProceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020
PublisherIEEE Computer Society
Pages589-600
Number of pages12
ISBN (Electronic)9781728196213
DOIs
StatePublished - Nov 2020
Externally publishedYes
Event61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States
Duration: 16 Nov 202019 Nov 2020

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2020-November
ISSN (Print)0272-5428

Conference

Conference61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020
Country/TerritoryUnited States
CityVirtual, Durham
Period16/11/2019/11/20

Bibliographical note

Publisher Copyright:
© 2020 IEEE.

Keywords

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

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