Advertisement

A Polynomial-Time Approximation Algorithm for a Geometric Dispersion Problem

  • Marc Benkert
  • Joachim Gudmundsson
  • Christian Knauer
  • Esther Moet
  • René van Oostrum
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)

Abstract

We consider the problem of placing a maximal number of disks in a rectangular region containing obstacles such that no two disks intersect. Let α be a fixed real in (0,1]. We are given a bounding rectangle P and a set \(\cal{R}\) of possibly intersecting unit disks whose centers lie in P. The task is to pack a set \(\cal{B}\) of m disjoint disks of radius α into P such that no disk in \(\cal{B}\) intersects a disk in \(\cal{R}\), where m is the maximum number of unit disks that can be packed. Baur and Fekete showed that the problem cannot be solved in polynomial time for α≥13/14, unless \({\cal P}={\cal NP}\). In this paper we present an algorithm for α= 2/3.

Keywords

Polynomial Time Unit Disk Pairwise Disjoint Original Optimization Problem Equal Circle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benkert, M., Gudmundsson, J., Knauer, C., Moet, E., van Oostrum, R., Wolff, A.: A polynomial-time approximation algorithm for a geometric dispersion problem. Technical Report 2005-8, Universität Karlsruhe (May 2006), available at: http://www.ubka.uni-karlsruhe.de/indexer-vvv/ira/2006/8
  2. 2.
    Baur, C., Fekete, S.P.: Approximation of Geometric Dispersion Problems. Algorithmica 30, 450–470 (2001)MathSciNetGoogle Scholar
  3. 3.
    Cappanera, P.: A survey on obnoxious facility location problems. Technical Report TR-99-11, University of Pisa (1999)Google Scholar
  4. 4.
    Chang, E.-C., Choi, S.W., Kwon, D., Park, H., Yap, C.-K.: Shortest path amidst disc obstacles is computable. In: Proceedings of the 21st ACM Symposium on Computational Geometry (2005)Google Scholar
  5. 5.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-compete. Information Processing Letters 12, 133–137 (1981)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Füredi, Z.: The densest packing of equal circles into a parallel strip. Discrete & Computational Geometry 6, 95–106 (1991)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hochbaum, D., Maas, W.: Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. Journal of the ACM 32, 130–136 (1985)MATHCrossRefGoogle Scholar
  8. 8.
    Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion admidst polygonal obstacles. Discrete & Computational Geometry 1, 59–71 (1986)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Maranas, C., Floudas, C., Pardalos, P.: New results in the packing of equal circles in a square. Discrete Mathematics 128, 187–293 (1995)MathSciNetGoogle Scholar
  10. 10.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM 30(4), 852–865 (1983)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rogers, C.A.: Packing and Covering. Cambridge University Press, Cambridge (1964)MATHGoogle Scholar
  12. 12.
    Tóth, G.F.: Packing and Covering. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., CRC Press LLC, Boca Raton (2004)Google Scholar
  13. 13.
    Zong, C., Talbot, J.: Sphere Packings. Springer, Heidelberg (1999)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Marc Benkert
    • 1
  • Joachim Gudmundsson
    • 2
  • Christian Knauer
    • 3
  • Esther Moet
    • 4
  • René van Oostrum
    • 4
  • Alexander Wolff
    • 1
  1. 1.Faculty of Computer ScienceKarlsruhe UniversityKarlsruheGermany
  2. 2.National ICT Australia LtdSydneyAustralia
  3. 3.Institute of Computer ScienceFreie Universität BerlinGermany
  4. 4.Department of Computing and Information SciencesUniversiteit UtrechtThe Netherlands

Personalised recommendations