Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Becker, Amariah; Klein, Philip N.; Saulpic, David http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-94710
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Polynomial-Time Approximation Schemes for k-center, k-median, and Capacitated Vehicle Routing in Bounded Highway Dimension

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Abstract

The concept of bounded highway dimension was developed to capture observed properties of road networks. We show that a graph of bounded highway dimension with a distinguished root vertex can be embedded into a graph of bounded treewidth in such a way that u-to-v distance is preserved up to an additive error of epsilon times the u-to-root plus v-to-root distances. We show that this embedding yields a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension. In this problem, the input specifies a depot and a set of clients, each with a location and demand; the output is a set of depot-to-depot tours, where each client is visited by some tour and each tour covers at most Q units of client demand. Our PTAS can be extended to handle penalties for unvisited clients. We extend this embedding result to handle a set S of root vertices. This result implies a PTAS for Multiple Depot Bounded-Capacity Vehicle Routing: the tours can go from one depot to another. The embedding result also implies that, for fixed k, there is a PTAS for k-Center in graphs of bounded highway dimension. In this problem, the goal is to minimize d so that there exist k vertices (the centers) such that every vertex is within distance d of some center. Similarly, for fixed k, there is a PTAS for k-Median in graphs of bounded highway dimension. In this problem, the goal is to minimize the sum of distances to the k centers.

BibTeX - Entry

@InProceedings{becker_et_al:LIPIcs:2018:9471,
  author =	{Amariah Becker and Philip N. Klein and David Saulpic},
  title =	{{Polynomial-Time Approximation Schemes for k-center, k-median, and Capacitated Vehicle Routing in Bounded Highway Dimension}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{8:1--8:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Yossi Azar and Hannah Bast and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9471},
  URN =		{urn:nbn:de:0030-drops-94710},
  doi =		{10.4230/LIPIcs.ESA.2018.8},
  annote =	{Keywords: Highway Dimension, Capacitated Vehicle Routing, Graph Embeddings}
}

Keywords: Highway Dimension, Capacitated Vehicle Routing, Graph Embeddings
Seminar: 26th Annual European Symposium on Algorithms (ESA 2018)
Issue date: 2018
Date of publication: 2018


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