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

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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).

BibTeX - Entry

@InProceedings{bandyapadhyay_et_al:LIPIcs:2018:9619,
  author =	{Sayan Bandyapadhyay and Anil Maheshwari and Saeed Mehrabi and Subhash Suri},
  title =	{{Approximating Dominating Set on Intersection Graphs of Rectangles and L-frames}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9619},
  URN =		{urn:nbn:de:0030-drops-96198},
  doi =		{10.4230/LIPIcs.MFCS.2018.37},
  annote =	{Keywords: Minimum dominating set, Rectangles and L-frames, Approximation schemes, Local search, APX-hardness}
}

Keywords: Minimum dominating set, Rectangles and L-frames, Approximation schemes, Local search, APX-hardness
Seminar: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue date: 2018
Date of publication: 2018


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