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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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Author Details

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

Funding

Research of Sayan Bandyapadhyay and Subhash Suri was supported in part by the NSF grant CCF-1525817. Research of Anil Maheshwari is supported in part by NSERC. Saeed Mehrabi is supported by a Carleton-Fields postdoctoral fellowship.
  • Bandyapadhyay, Sayan: The work was partially done when the author was visiting University of California, Santa Barbara.

Cite AsGet BibTex

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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