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DOI: 10.4230/LIPIcs.SWAT.2018.19
URN: urn:nbn:de:0030-drops-88450
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8845/
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Feldmann, Andreas Emil ; Marx, Dániel

The Parameterized Hardness of the k-Center Problem in Transportation Networks

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LIPIcs-SWAT-2018-19.pdf (0.5 MB)


Abstract

In this paper we study the hardness of the k-Center problem on inputs that model transportation networks. For the problem, an edge-weighted graph G=(V,E) and an integer k are given and a center set C subseteq V needs to be chosen such that |C|<= k. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the k-Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and even the treewidth t. Moreover, under the Exponential Time Hypothesis there is no f(k,t,h)* n^{o(t+sqrt{k+h})} time algorithm for any computable function f. Thus it is unlikely that the optimum solution to k-Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized (1+{epsilon})-approximation algorithm for inputs of doubling dimension d with runtime (k^k/{epsilon}^{O(kd)})* n^{O(1)}. This generalizes a previous result, which considered inputs in D-dimensional L_q metrics.

BibTeX - Entry

@InProceedings{feldmann_et_al:LIPIcs:2018:8845,
  author =	{Andreas Emil Feldmann and D{\'a}niel Marx},
  title =	{{The Parameterized Hardness of the k-Center Problem in Transportation Networks}},
  booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm  Theory (SWAT 2018)},
  pages =	{19:1--19:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-068-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{101},
  editor =	{David Eppstein},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8845},
  URN =		{urn:nbn:de:0030-drops-88450},
  doi =		{10.4230/LIPIcs.SWAT.2018.19},
  annote =	{Keywords: k-center, parameterized complexity, planar graphs, doubling dimension, highway dimension, treewidth}
}

Keywords: k-center, parameterized complexity, planar graphs, doubling dimension, highway dimension, treewidth
Seminar: 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
Issue Date: 2018
Date of publication: 30.05.2018


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