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DOI: 10.4230/LIPIcs.SoCG.2018.2
URN: urn:nbn:de:0030-drops-87152
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8715/
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Abu-Affash, A. Karim ; Carmi, Paz ; Maheshwari, Anil ; Morin, Pat ; Smid, Michiel ; Smorodinsky, Shakhar

Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

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LIPIcs-SoCG-2018-2.pdf (0.6 MB)


Abstract

We consider a well studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and an integer d >= 1, in the maximum diameter-bounded subgraph problem (MaxDBS for short), the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For d=1, this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor n^{1-epsilon}, for any epsilon > 0. Moreover, it is known that, for any d >= 2, it is NP-hard to approximate MaxDBS within a factor n^{1/2 - epsilon}, for any epsilon > 0. In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems and several geometric properties of unit disk graphs.

BibTeX - Entry

@InProceedings{abuaffash_et_al:LIPIcs:2018:8715,
  author =	{A. Karim Abu-Affash and Paz Carmi and Anil Maheshwari and Pat Morin and Michiel Smid and Shakhar Smorodinsky},
  title =	{{Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{2:1--2:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8715},
  URN =		{urn:nbn:de:0030-drops-87152},
  doi =		{10.4230/LIPIcs.SoCG.2018.2},
  annote =	{Keywords: Approximation algorithms, maximum diameter-bounded subgraph, unit disk graphs, fractional Helly theorem, VC-dimension}
}

Keywords: Approximation algorithms, maximum diameter-bounded subgraph, unit disk graphs, fractional Helly theorem, VC-dimension
Seminar: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 24.05.2018


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