Efficient Approximation Schemes for Geometric Problems?

  • Dániel Marx
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


An EPTAS (efficient PTAS) is an approximation scheme where ε does not appear in the exponent of n, i.e., the running time is f(ε) Open image in new window n c . We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an open question of Alber and Fiala [2], we show that Maximum Independent Set is W[1]-complete for the intersection graphs of unit disks and axis-parallel unit squares in the plane. A standard consequence of this result is that the \(n^{O(1/{\it \epsilon})}\) time PTAS of Hunt et al. [11] for Maximum Independent Set on unit disk graphs cannot be improved to an EPTAS. Similar results are obtained for the problem of covering points with squares.


Unit Disk Approximation Scheme Test Point Vertex Cover Maximum Clique 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. 11(3-4), 209–218 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alber, J., Fiala, J.: Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms 52(2), 134–151 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arora, S.: Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In: FOCS 1996, pp. 2–11. IEEE Comput. Soc. Press, Los Alamitos (1996)Google Scholar
  4. 4.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bazgan, C.: Schémas d’approximation et complexité paramétrée. Technical report, Université Paris Sud (1995)Google Scholar
  6. 6.
    Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Inform. Process. Lett. 64(4), 165–171 (1997)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Downey, R.G.: Parameterized complexity for the skeptic. In: Proceedings of the 18th IEEE Annual Conference on Computational Complexity, pp. 147–169 (2003)Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. In: Monographs in Computer Science. Springer, New York (1999)Google Scholar
  9. 9.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric graphs. In: SODA 2001, pp. 671–679. SIAM, Philadelphia (2001)Google Scholar
  10. 10.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26(2), 238–274 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Malesińska, E.: Graph-Thoretical Models for Frequency Assignment Problems. PhD thesis, Technical University of Berlin (1997)Google Scholar
  13. 13.
    Sunil Chandran, L., Grandoni, F.: Refined memorization for vertex cover. Inform. Process. Lett. 93(3), 125–131 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Wang, D.W., Kuo, Y.-S.: A study on two geometric location problems. Inform. Process. Lett. 28(6), 281–286 (1988)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dániel Marx
    • 1
  1. 1.Department of Computer Science and Information TheoryBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations