Skip to main content
SpringerLink
Log in

On the Clique Problem in Intersection Graphs of Ellipses

  • Conference paper
  • First Online:
Algorithms and Computation (ISAAC 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2518))

Included in the following conference series:

Abstract

Intersection graphs of disks and of line segments, respectively, have been well studied, because of both, practical applications and theoretically interesting properties of these graphs. Despite partial results, the complexity status of the Clique problem for these two graph classes is still open.

Here, we consider the Clique problem for intersection graphs of ellipses which in a sense, interpolate between disc and ellipses, and show that it is \( \mathcal{A}\mathcal{P}\mathcal{X} \)-hard in that case. Moreover, this holds even if for all ellipses, the ratio of the larger over the smaller radius is some prescribed number.

To our knowledge, this is the first hardness result for the Clique problem in intersection graphs of objects with finite description complexity. We also describe a simple approximation algorithm for the case of ellipses for which the ratio of radii is bounded.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. I. Aardal, S.P. M. van Hoesel, A. M. C. A. Kosterand C. Mannino, and A. Sassano. Models and solution techniques for frequency assignment problems. Technical report, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 2001.

    Google Scholar 

  2. N. Alon and D. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem. Advances in Mathematics, 96(1):103–112, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Alon, H. Last, R. Pinchasi, and M. Sharir. On the complexity of arrangements of circles in the plane. Discrete Comput. Geom., 26(4):465–492, 2001.

    MATH  MathSciNet  Google Scholar 

  4. S. Arora and C. Lund. Hardness of approximations. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS, 1995.

    Google Scholar 

  5. C. J. Colbourn B. N. Clark and D. S. Johnson. Unit disk graphs. Discrete Mathematics, 86:165–177, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Brandstädt, V. B. Le, and J. P. Spinrad. Graph classes: a survey. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.

    MATH  Google Scholar 

  7. H. Breu and D. G. Kirkpatrick. Unit disk graph recognition is NP-hard. Computational Geometry, 9(1–2):3–24, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Danzer. Zur Lösung des Gallaischen Problems über Kreisscheiben in der euklidischen Ebene. Studia Scientiarum Mathematicorum Hungarica, 21:111–134, 1986.

    MATH  MathSciNet  Google Scholar 

  9. T. Erlebach, K. Jansen, and E. Seidel. Polynomial-time approximation schemes for geometric graphs. In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA’01), pages 671–679, 2001.

    Google Scholar 

  10. P. Hliněný and J. Kratochvíl. Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Math., 229(1–3): 101–124, 2001. Combinatorics, graph theory, algorithms and applications.

    Article  MathSciNet  MATH  Google Scholar 

  11. Petr Hliněný. Contact graphs of line segments are NP-complete. Discrete Math., 235(1–3):95–106, 2001. Combinatorics (Prague, 1998).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Koebe. Kontaktprobleme der konformen Abbildung. Berichte uber die Verhandlungen der Sachsischen Akademie der Wissenschaften, Math.-Phys. Klasse, 88:141–164, 1936.

    Google Scholar 

  13. J. Kratochvíl and A. Kuběna. On intersection representations of co-planar graphs. Discrete Math., 178(1–3):251–255, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Kratochvíl and J. Matoušek. Intersection graphs of segments. J. Combin. Theory Ser. B, 62(2):289–315, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  15. T. A. McKee and F. R. McMorris. Topics in intersection graph theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.

    MATH  Google Scholar 

  16. M. Sharir and P.K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge, UK, 1995.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Informatik, ETH Zürich ETH Zentrum, 8092, Zürich, Switzerland

    Christoph Ambühl & Uli Wagner

Authors
  1. Christoph Ambühl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uli Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada, K1S 5B6

    Prosenjit Bose  & Pat Morin  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ambühl, C., Wagner, U. (2002). On the Clique Problem in Intersection Graphs of Ellipses. In: Bose, P., Morin, P. (eds) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36136-7_43

Download citation

  • DOI: https://doi.org/10.1007/3-540-36136-7_43

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00142-3

  • Online ISBN: 978-3-540-36136-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation