Linear Problem Kernels for NP-Hard Problems on Planar Graphs

  • Jiong Guo
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, and Efficient Dominating Set on planar graphs. On the route to these results, we present effective, problem-specific data reduction rules that are useful in any approach attacking the computational intractability of these problems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the Vertex Cover problem: theory and experiments. In: Proc. 6th ACM-SIAM ALENEX, pp. 62–69. ACM Press, New York (2004)Google Scholar
  2. 2.
    Alber, J., Betzler, N., Niedermeier, R.: Experiments on data reduction for optimal domination in networks. Annals of Operations Research 146(1), 105–117 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alber, J., Dorn, B., Niedermeier, R.: A general data reduction scheme for domination in graphs. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 137–147. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial time data reduction for Dominating Set. Journal of the ACM 51(3), 363–384 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41(1), 153–180 (1994)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bange, D.W., Barkauskas, A.E., Slater, P.J.: Efficient dominating sets in graphs. In: Proc. 3rd Conference on Discrete Mathematics. SIAM, pp. 189–199 (1988)Google Scholar
  7. 7.
    Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 269–280. Springer, Heidelberg (2005)Google Scholar
  8. 8.
    Chen, J., Kanj, I.A., Jia, W.: Vertex Cover: Further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Fellows, M.R., Heggernes, P., Rosamond, F.A., Sloper, C., Telle, J.A.: Finding k disjoint triangles in an arbitrary graph. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 235–244. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    Fellows, M.R., Hoover, M.: Perfect domination. Australian Journal of Combinatorics 3, 141–150 (1991)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fomin, F.V., Thilikos, D.M.: Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speed-up. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 581–592. Springer, Heidelberg (2004)Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM Journal on Applied Mathematics 32, 826–834 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  16. 16.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of generalized Vertex Cover problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 36–48. Springer, Heidelberg (2005) Long version to appear under the title Parameterized complexity of Vertex Cover variants in Theory of Computing SystemsGoogle Scholar
  17. 17.
    Guo, J., Niedermeier, R., Wernicke, S.: Fixed-parameter tractability results for full-degree spanning tree and its dual. In: Fischer, K., Timm, I.J., André, E., Zhong, N. (eds.) MATES 2006. LNCS (LNAI), vol. 4196, pp. 203–214. Springer, Heidelberg (2006)Google Scholar
  18. 18.
    Lu, C.L., Tang, C.Y.: Weighted efficient domination problem on some perfect graphs. Discrete Applied Mathematics 117, 163–182 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Moser, H., Sikdar, S.: The parameterized complexity of the induced matching problem in planar graphs. In: FAW 2007. Proc. 1st International Frontiers of Algorithmics Workshop. LNCS, Springer, Heidelberg (2007)Google Scholar
  20. 20.
    Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  22. 22.
    Prieto, E.: Systematic Kernelization in FPT Algorithm Design. PhD thesis, Department of Computer Science, University of Newcastle, Australia (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jiong Guo
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 JenaGermany

Personalised recommendations