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Approximating Dominating Set on Intersection Graphs of Rectangles and L-frames

Authors Sayan Bandyapadhyay, Anil Maheshwari, Saeed Mehrabi, Subhash Suri



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Sayan Bandyapadhyay
  • Department of Computer Science, University of Iowa, Iowa City, USA
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Saeed Mehrabi
  • School of Computer Science, Carleton University, Ottawa, Canada
Subhash Suri
  • Department of Computer Science, UC Santa Barbara, California, USA

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Sayan Bandyapadhyay, Anil Maheshwari, Saeed Mehrabi, and Subhash Suri. Approximating Dominating Set on Intersection Graphs of Rectangles and L-frames. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 37:1-37:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.37

Abstract

We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sub-logarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any epsilon>0, there exists a (2+epsilon)-approximation algorithm for the MDS problem on "diagonal-anchored" rectangles, providing the first O(1)-approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on "diagonal-anchored" rectangles is the same as the MDS problem on "diagonal-anchored" L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a (2+epsilon)-approximation for the problem with "diagonal-anchored" L-frames. On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017).

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Computational geometry
Keywords
  • Minimum dominating set
  • Rectangles and L-frames
  • Approximation schemes
  • Local search
  • APX-hardness

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