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Efficient Approximation Schemes for Geometric Problems?

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Algorithms – ESA 2005 (ESA 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3669))

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Abstract

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) 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.

Research is supported in part by grants OTKA 44733, 42559 and 42706 of the Hungarian National Science Fund.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bazgan, C.: Schémas d’approximation et complexité paramétrée. Technical report, Université Paris Sud (1995)

    Google Scholar 

  6. Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Inform. Process. Lett. 64(4), 165–171 (1997)

    Article  MathSciNet  Google Scholar 

  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. Downey, R.G., Fellows, M.R.: Parameterized complexity. In: Monographs in Computer Science. Springer, New York (1999)

    Google Scholar 

  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. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Malesińska, E.: Graph-Thoretical Models for Frequency Assignment Problems. PhD thesis, Technical University of Berlin (1997)

    Google Scholar 

  13. Sunil Chandran, L., Grandoni, F.: Refined memorization for vertex cover. Inform. Process. Lett. 93(3), 125–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wang, D.W., Kuo, Y.-S.: A study on two geometric location problems. Inform. Process. Lett. 28(6), 281–286 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest, H-1521, Hungary

    Dániel Marx

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  1. Dániel Marx
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Editors and Affiliations

  1. BRICS, MADALGO, Department of Computer Science, University of Aarhus, Denmark

    Gerth Stølting Brodal

  2. University of Rome “La Sapienza”, Rome, Italy

    Stefano Leonardi

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© 2005 Springer-Verlag Berlin Heidelberg

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Marx, D. (2005). Efficient Approximation Schemes for Geometric Problems?. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_41

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  • DOI: https://doi.org/10.1007/11561071_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29118-3

  • Online ISBN: 978-3-540-31951-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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