Low Treewidth Embeddings of Planar and Minor-Free Metrics

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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 languageEnglish
Title of host publicationProceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022
PublisherIEEE Computer Society
Pages1081-1092
Number of pages12
ISBN (Electronic)9781665455190
DOIs
StatePublished - 2022
Event63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 - Denver, United States
Duration: 31 Oct 20223 Nov 2022

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2022-October
ISSN (Print)0272-5428

Conference

Conference63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022
Country/TerritoryUnited States
CityDenver
Period31/10/223/11/22

Bibliographical note

Publisher Copyright:
© 2022 IEEE.

Keywords

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

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