Improved Local Search for Geometric Hitting Set

Authors Norbert Bus, Shashwat Garg, Nabil H. Mustafa, Saurabh Ray



PDF
Thumbnail PDF

File

LIPIcs.STACS.2015.184.pdf
  • Filesize: 0.76 MB
  • 13 pages

Document Identifiers

Author Details

Norbert Bus
Shashwat Garg
Nabil H. Mustafa
Saurabh Ray

Cite AsGet BibTex

Norbert Bus, Shashwat Garg, Nabil H. Mustafa, and Saurabh Ray. Improved Local Search for Geometric Hitting Set. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 184-196, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.STACS.2015.184

Abstract

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3,2)-local search and give an (8+\epsilon)-approximation algorithm with expected running time ˜O(n^{2.34}); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n^{15}) -- that too just for unit disks. The techniques and ideas generalize to (4,3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3,2) local search gives an 8-approximation and no better \footnote{This is assuming the use of the standard framework. Improvement of the approximation factor by using additional properties specific to the problem may be possible.}. Similarly (4,3)-local search gives a 5-approximation for all these problems.
Keywords
  • hitting sets
  • Delaunay triangulation
  • local search
  • disks
  • geometric algorithms

Metrics

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

References

  1. Rashmisnata Acharyya, Manjanna Basappa, and Gautam K. Das. Unit disk cover problem in 2D. In Computational Science and Its Applications - ICCSA 2013 - 13th International Conference, Ho Chi Minh City, Vietnam, June 24-27, 2013, Proceedings, Part II, pages 73-85, 2013. Google Scholar
  2. Peyman Afshani and Timothy M. Chan. Optimal halfspace range reporting in three dimensions. In SODA, pages 180-186, 2009. Google Scholar
  3. Pankaj K. Agarwal, Esther Ezra, and Micha Sharir. Near-linear approximation algorithms for geometric hitting sets. Algorithmica, 63(1-2):1-25, 2012. Google Scholar
  4. Pankaj K. Agarwal and Nabil H. Mustafa. Independent set of intersection graphs of convex objects in 2D. Comput. Geom., 34(2):83-95, 2006. Google Scholar
  5. Pankaj K. Agarwal and Jiangwei Pan. Near-linear algorithms for geometric hitting sets and set covers. In Symposium on Computational Geometry, page 271, 2014. Google Scholar
  6. Christoph Ambühl, Thomas Erlebach, Matús Mihalák, and Marc Nunkesser. Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In APPROX-RANDOM, pages 3-14, 2006. Google Scholar
  7. Boris Aronov and Sariel Har-Peled. On approximating the depth and related problems. SIAM J. Comput., 38(3):899-921, 2008. Google Scholar
  8. H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete & Computational Geometry, 14(4):463-479, 1995. Google Scholar
  9. Gruia Călinescu, Ion I. Mandoiu, Peng-Jun Wan, and Alexander Zelikovsky. Selecting forwarding neighbors in wireless ad hoc networks. MONET, 9(2):101-111, 2004. Google Scholar
  10. P. Carmi, M. Katz, and N. Lev-Tov. Covering points by unit disks of fixed location. In ISAAC, pages 644-655, 2007. Google Scholar
  11. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. In Symposium on Computational Geometry, pages 333-340, 2009. Google Scholar
  12. Francisco Claude, Gautam K. Das, Reza Dorrigiv, Stephane Durocher, Robert Fraser, Alejandro López-Ortiz, Bradford G. Nickerson, and Alejandro Salinger. An improved line-separable algorithm for discrete unit disk cover. Discrete Math., Alg. and Appl., 2(1):77-88, 2010. Google Scholar
  13. Gautam K. Das, Robert Fraser, Alejandro LÃsuperscript3pez-Ortiz, and Bradford G. Nickerson. On the discrete unit disk cover problem. International Journal on Computational Geometry and Applications, 22(5):407-419, 2012. Google Scholar
  14. Michael B. Dillencourt and Warren D. Smith. Graph-theoretical conditions for inscribability and delaunay realizability. Discrete Mathematics, 161(1–3):63-77, 1996. Google Scholar
  15. G. Even, D. Rawitz, and S. Shahar. Hitting sets when the VC-dimension is small. Inf. Process. Lett., 95:358-362, 2005. Google Scholar
  16. Robert Fraser. Algorithms for Geometric Covering and Piercing Problems. PhD thesis, University of Waterloo, 2012. Google Scholar
  17. Matt Gibson and Imran A. Pirwani. Algorithms for dominating set in disk graphs: Breaking the logn barrier. In ESA, pages 243-254, 2010. Google Scholar
  18. Dorit S. Hochbaum and Wolfgang Maass. Fast approximation algorithms for a nonconvex covering problem. J. Algorithms, 8(3):305-323, 1987. Google Scholar
  19. Erik Krohn, Matt Gibson, Gaurav Kanade, and Kasturi R. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168-178, 2014. Google Scholar
  20. 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
  21. Jiri Matousek. Lectures in Discrete Geometry. Springer-Verlag, New York, NY, 2002. Google Scholar
  22. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. Google Scholar
  23. Evangalia Pyrga and Saurabh Ray. New existence proofs for epsilon-nets. In Proceedings of Symposium on Computational Geometry, pages 199-207, 2008. Google Scholar
  24. Ran Raz and Muli Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In STOC, pages 475-484, 1997. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail