Approximating maximum diameter-bounded subgraph in unit disk graphs

A. Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, Shakhar Smorodinsky

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

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 n1-ϵ, for any ϵ > 0. Moreover, it is known that, for any d ≥ 2, it is NP-hard to approximate MaxDBS within a factor n1/2-ϵ, for any ϵ > 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.

Original languageEnglish
Title of host publication34th International Symposium on Computational Geometry, SoCG 2018
EditorsCsaba D. Toth, Bettina Speckmann
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages21-212
Number of pages192
ISBN (Electronic)9783959770668
DOIs
StatePublished - 1 Jun 2018
Event34th International Symposium on Computational Geometry, SoCG 2018 - Budapest, Hungary
Duration: 11 Jun 201814 Jun 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume99
ISSN (Print)1868-8969

Conference

Conference34th International Symposium on Computational Geometry, SoCG 2018
Country/TerritoryHungary
CityBudapest
Period11/06/1814/06/18

Keywords

  • Approximation algorithms
  • Fractional helly theorem
  • Maximum diameter-bounded subgraph
  • Unit disk graphs
  • VC-dimension

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