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DOI: 10.4230/LIPIcs.STACS.2015.184
URN: urn:nbn:de:0030-drops-49135
URL: http://drops.dagstuhl.de/opus/volltexte/2015/4913/
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Bus, Norbert ; Garg, Shashwat ; Mustafa, Nabil H. ; Ray, Saurabh

Improved Local Search for Geometric Hitting Set

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Abstract

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3,2)-local search and give an (8+\epsilon)-approximation algorithm with expected running time O(n^{2.34}); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n^{15}) -- that too just for unit disks. The techniques and ideas generalize to (4,3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3,2) local search gives an 8-approximation and no better \footnote{This is assuming the use of the standard framework. Improvement of the approximation factor by using additional properties specific to the problem may be possible.}. Similarly (4,3)-local search gives a 5-approximation for all these problems.

BibTeX - Entry

@InProceedings{bus_et_al:LIPIcs:2015:4913,
  author =	{Norbert Bus and Shashwat Garg and Nabil H. Mustafa and Saurabh Ray},
  title =	{{Improved Local Search for Geometric Hitting Set}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{184--196},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Ernst W. Mayr and Nicolas Ollinger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/4913},
  URN =		{urn:nbn:de:0030-drops-49135},
  doi =		{10.4230/LIPIcs.STACS.2015.184},
  annote =	{Keywords: hitting sets, Delaunay triangulation, local search, disks, geometric algorithms}
}

Keywords: hitting sets, Delaunay triangulation, local search, disks, geometric algorithms
Seminar: 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)
Issue Date: 2015
Date of publication: 25.02.2015


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