Document Open Access Logo

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

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

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2018.37.pdf
  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Andrei Asinowski, Elad Cohen, Martin Charles Golumbic, Vincent Limouzy, Marina Lipshteyn, and Michal Stern. Vertex intersection graphs of paths on a grid. J. Graph Algorithms Appl., 16(2):129-150, 2012. Google Scholar
  2. Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The hardness of approximation of Euclidean k-means. In SoCG 2015, Netherlands, 754-767, volume 34 of Leibniz International Proceedings in Informatics (LIPIcs), 2015. Google Scholar
  3. Sayan Bandyapadhyay, Anil Maheshwari, Saeed Mehrabi, and Subhash Suri. Approximating dominating set on intersection graphs of L-frames. CoRR, abs/1803.06216, 2018. Google Scholar
  4. Ayelet Butman, Danny Hermelin, Moshe Lewenstein, and Dror Rawitz. Optimization problems in multiple-interval graphs. ACM Trans. Algorithms, 6(2):40:1-40:18, 2010. Google Scholar
  5. Daniele Catanzaro, Steven Chaplick, Stefan Felsner, Bjarni V. Halldórsson, Magnús M. Halldórsson, Thomas Hixon, and Juraj Stacho. Max point-tolerance graphs. Discrete Applied Mathematics, 216:84-97, 2017. Google Scholar
  6. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. Google Scholar
  7. H. S. Chao, Fang-Rong Hsu, and Richard C. T. Lee. An optimal algorithm for finding the minimum cardinality dominating set on permutation graphs. Discrete Applied Mathematics, 102(3):159-173, 2000. Google Scholar
  8. Victor Chepoi and Stefan Felsner. Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve. Comput. Geom., 46(9):1036-1041, 2013. Google Scholar
  9. José R. Correa, Laurent Feuilloley, Pablo Pérez-Lantero, and José A. Soto. Independent and hitting sets of rectangles intersecting a diagonal line: Algorithms and complexity. DCG, 53(2):344-365, 2015. Google Scholar
  10. Mirela Damian and Sriram V. Pemmaraju. APX-hardness of domination problems in circle graphs. Inf. Process. Lett., 97(6):231-237, 2006. Google Scholar
  11. Mirela Damian-Iordache and Sriram V. Pemmaraju. A (2+epsilon)-approximation scheme for minimum domination on circle graphs. J. Algorithms, 42(2):255-276, 2002. Google Scholar
  12. Minati De and Abhiruk Lahiri. Geometric dominating set and set cover via local search. CoRR, abs/1605.02499, 2016. URL: http://arxiv.org/abs/1605.02499.
  13. Mark de Berg and Amirali Khosravi. Optimal binary space partitions for segments in the plane. Int. J. Comput. Geometry Appl., 22(3):187-206, 2012. Google Scholar
  14. Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, 162(1):439-485, 2005. Google Scholar
  15. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 624-633, 2014. Google Scholar
  16. Thomas Erlebach and Erik Jan van Leeuwen. Domination in geometric intersection graphs. In LATIN 2008, Búzios, Brazil, April 7-11, 2008, Proceedings, pages 747-758, 2008. Google Scholar
  17. Matt Gibson and Imran A. Pirwani. Algorithms for dominating set in disk graphs: Breaking the logn barrier - (extended abstract). In ESA 2010, UK, pages 243-254, 2010. Google Scholar
  18. Martin Charles Golumbic, Marina Lipshteyn, and Michal Stern. Edge intersection graphs of single bend paths on a grid. Networks, 54(3):130-138, 2009. Google Scholar
  19. Sathish Govindarajan, Rajiv Raman, Saurabh Ray, and Aniket Basu Roy. Packing and covering with non-piercing regions. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 47:1-47:17, 2016. Google Scholar
  20. Daniel Heldt, Kolja B. Knauer, and Torsten Ueckerdt. Edge-intersection graphs of grid paths: The bend-number. Discrete Applied Mathematics, 167:144-162, 2014. Google Scholar
  21. Joseph D. Horton and Kyriakos Kilakos. Minimum edge dominating sets. SIAM J. Discrete Math., 6(3):375-387, 1993. Google Scholar
  22. J. Mark Keil, Joseph S. B. Mitchell, Dinabandhu Pradhan, and Martin Vatshelle. An algorithm for the maximum weight independent set problem on outerstring graphs. Comput. Geom., 60:19-25, 2017. Google Scholar
  23. Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 154-165, 2006. Google Scholar
  24. Saeed Mehrabi. Approximating domination on intersection graphs of paths on a grid. In 15th International Workshop on Approximation and Online Algorithms (WAOA 2017), Vienna, Austria, pages 76-89, 2017. Google Scholar
  25. Apurva Mudgal and Supantha Pandit. Covering, hitting, piercing and packing rectangles intersecting an inclined line. In COCOA 2015, Houston, TX, USA, pages 126-137, 2015. Google Scholar
  26. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. Google Scholar
  27. Supantha Pandit. Dominating set of rectangles intersecting a straight line. In CCCG 2017, Ottawa, Ontario, Canada, pages 144-149, 2017. Google Scholar
  28. Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 475-484, 1997. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact