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Approximation Schemes for Geometric Coverage Problems

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

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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
  • and , Institute of Computer Science, University of Wrocław, Poland

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Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase. Approximation Schemes for Geometric Coverage Problems. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 17:1-17:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


In their seminal work, Mustafa and Ray [Nabil H. Mustafa and Saurabh Ray, 2010] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search - this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [Greg N. Frederickson, 1987]. Obtaining similar results for the related maximum coverage problem (MC) seems non-trivial due to the hard cardinality constraint. In fact, while Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] have shown (via a different analysis) that local search yields a PTAS for two-dimensional real halfspaces, they only conjectured that the same holds true for dimension three. Interestingly, at this point it was already known that local search provides a PTAS for the corresponding set cover case and this followed directly from the approach of Mustafa and Ray. In this work we provide a way to address the above-mentioned issue. First, we propose a color-balanced version of the planar separator theorem. The resulting subdivision approximates locally in each part the global distribution of the colors. Second, we show how this roughly balanced subdivision can be employed in a more careful analysis to strictly obey the hard cardinality constraint. More specifically, we obtain a PTAS for any "planarizable" instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we confirm the conjecture of Badanidiyuru, Kleinberg, and Lee [Ashwinkumar Badanidiyuru et al., 2012] regarding real halfspaces in dimension three. We feel that our ideas could also be helpful in other geometric settings involving a cardinality constraint.

Subject Classification

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


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