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Brief Announcement: Approximation Schemes for Geometric Coverage Problems

Authors Steven Chaplick , Minati De, Alexander Ravsky, Joachim Spoerhase

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ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • balanced separators
  • maximum coverage
  • local search
  • approximation scheme
  • geometric approximation algorithms

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Abstract

In this announcement, we show that the classical Maximum Coverage problem (MC) admits a PTAS via local search in essentially all cases where the corresponding instances of Set Cover (SC) admit a PTAS via the local search approach by Mustafa and Ray [Nabil H. Mustafa and Saurabh Ray, 2010]. As a corollary, we answer an open question by Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] regarding half-spaces in R^3 thereby settling the existence of PTASs for essentially all natural cases of geometric MC problems. As an intermediate result, we show a color-balanced version of the classical planar subdivision theorem by Frederickson [Greg N. Frederickson, 1987]. We believe that some of our ideas may be useful for analyzing local search in other settings involving a hard cardinality constraint.

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Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase. Brief Announcement: Approximation Schemes for Geometric Coverage Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 107:1-107:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.107

Author Details

Steven Chaplick
  • Lehrstuhl für Informatik I, Universität Würzburg, Germany
Minati De
  • Department of Mathematics, Indian Institute of Technology Delhi, India
Alexander Ravsky
  • Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Science of Ukraine, Lviv, Ukraine
Joachim Spoerhase
  • Lehrstuhl für Informatik I, Universität Würzburg, Germany
  • , Institute of Computer Science, University of Wrocław, Poland.

Funding

  • De, Minati: Partially supported by DST-INSPIRE Faculty Grant (DST-IFA-14-ENG-75)
  • Spoerhase, Joachim: Partially supported by Polish National Science Centre grant 2015/18/E/ST6/00456

References

  1. Ashwinkumar Badanidiyuru, Robert Kleinberg, and Hooyeon Lee. Approximating low-dimensional coverage problems. In Symp. Computational Geometry (SoCG'12), pages 161-170, 2012. URL: http://dx.doi.org/10.1145/2261250.2261274.
  2. Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase. Approximation schemes for geometric coverage problems. CoRR, 2016. URL: http://arxiv.org/abs/1607.06665.
  3. Gérard Cornuéjols, George L. Nemhauser, and Laurence A. Wolsey. Worst-case and probabilistic analysis of algorithms for a location problem. Operations Research, 28(4):847-858, 1980. URL: http://dx.doi.org/10.1287/opre.28.4.847.
  4. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: http://dx.doi.org/10.1145/285055.285059.
  5. Greg N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput., 16(6):1004-1022, 1987. Google Scholar
  6. Erik Krohn, Matt Gibson, Gaurav Kanade, and Kasturi R. Varadarajan. Guarding terrains via local search. J. of Computational Geometry, 5(1):168-178, 2014. URL: http://jocg.org/index.php/jocg/article/view/128.
  7. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. Google Scholar
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