License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2017.40
URN: urn:nbn:de:0030-drops-72289
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7228/
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Elbassioni, Khaled

Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension

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LIPIcs-SoCG-2017-40.pdf (0.6 MB)

Abstract

We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta).

BibTeX - Entry

@InProceedings{elbassioni:LIPIcs:2017:7228,
  author =	{Khaled Elbassioni},
  title =	{{Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{40:1--40:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7228},
  URN =		{urn:nbn:de:0030-drops-72289},
  doi =		{10.4230/LIPIcs.SoCG.2017.40},
  annote =	{Keywords: VC-dimension, approximation algorithms, fractional covering, multiplicative weights update, art gallery problem, polyhedral separators, geometric cove}
}

Keywords: VC-dimension, approximation algorithms, fractional covering, multiplicative weights update, art gallery problem, polyhedral separators, geometric cove
Collection: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 20.06.2017

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