Parameterized Complexity: Exponential Speed-Up for Planar Graph Problems

  • Jochen Alber
  • Henning Fernau
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2076)


A parameterized problem is fixed parameter tractable if it admits a solving algorithm whose running time on input instance (I,k) is f(k) · |I|;α, where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c k for a large variety of planar graph problems. The key to this type of algorithm is what we call the “Layerwise Separation Property” of a planar graph problem. Problems having this property include PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, and PLANAR DOMINATING SET.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Alber, H. Bodlaender, H. Fernau, and R. Niedermeier. Fixed parameter algorithms for planar dominating set and related problems. In Proc. 7th SWAT, vol. 1851 of LNCS, Springer, pp. 97–110, 2000. Full version available as Technical Report UU-CS-2000-28, Utrecht University, 2000.Google Scholar
  2. 2.
    J. Alber, H. Fernau, and R. Niedermeier. Parameterized complexity: exponential speed-up for planar graph problems. Technical Report TR01-023, ECCC Reports, Trier, March 2001. Available through
  3. 3.
    J. Alber, H. Fernau, and R. Niedermeier. Graph separators: a parameterized view. To appear in Proc. 7th COCOON, 2001. Full version available as Technical Report WSI-2001-8, Universität Tübingen (Germany), Wilhelm-Schickard-Institut Für Informatik, March 2001.Google Scholar
  4. 4.
    B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 41(1):153–180, 1994.zbMATHCrossRefGoogle Scholar
  5. 5.
    H. L. Bodlaender. Treewidth: Algorithmic techniques and results. In Proc. 22nd MFCS, vol. 1295 of LNCS, Springer, pp. 19–36, 1997.Google Scholar
  6. 6.
    H. L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sci., 209:1–45, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    L. Cai and D. Juedes. Subexponential parameterized algorithms collapse the Whierarchy. In Proc. 28th ICALP, 2001.Google Scholar
  8. 8.
    J. Chen, I. Kanj, and W. Jia. Vertex cover: Further observations and further improvements. In Proc. 25th WG, vol. 1665 of LNCS, Springer, pp. 313–324, 1999.Google Scholar
  9. 9.
    H. N. Djidjev and S. Venkatesan. Reduced constants for simple cycle graph separation. Acta Informatica, 34:231–243, 1997.CrossRefMathSciNetGoogle Scholar
  10. 10.
    R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer, 1999.Google Scholar
  11. 11.
    T. Kloks. Treewidth: Computations and Approximations, vol. 842 of LNCS, Springer, 1994.zbMATHGoogle Scholar
  12. 12.
    R. J. Lipton and R. E. Tarjan. Applications of a planar separator theorem. SIAM J. Comp., 9(3):615–627, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    K. Mehlhorn and S. Näher. LEDA: A Platform of Combinatorial and Geometric Computing. Cambridge University Press, Cambridge, England, 1999.zbMATHGoogle Scholar
  14. 14.
    R. Niedermeier and P. Rossmanith. Upper Bounds for Vertex Cover further improved. In Proc. 16th STACS, vol. 1563 of LNCS, Springer, pp. 561–570, 1999.CrossRefGoogle Scholar
  15. 15.
    G. L. Nemhauser and J. L. E. Trotter. Vertex packing: structural properties and algorithms. Math. Progr., 8:232–248, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    N. Robertson, D. P. Sanders, P. Seymour, and R. Thomas. Efficiently four-coloring planar graphs. In Proc. 28th STOC, ACM Press, pp. 571–575, 1996.Google Scholar
  17. 17.
    J. A. Telle and A. Proskurowski. Practical algorithms on partial k-trees with an application to domination-like problems. In Proc. 3rd WADS, vol. 709 of LNCS, Springer, pp. 610–621, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jochen Alber
    • 1
  • Henning Fernau
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany

Personalised recommendations