A Structural View on Parameterizing Problems: Distance from Triviality

  • Jiong Guo
  • Falk Hüffner
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3162)

Abstract

Based on a series of known and new examples, we propose the generalized setting of “distance from triviality” measurement as a reasonable and prospective way of determining useful structural problem parameters in analyzing computationally hard problems. The underlying idea is to consider tractable special cases of generally hard problems and to introduce parameters that measure the distance from these special cases. In this paper we present several case studies of distance from triviality parameterizations (concerning Clique, Power Dominating Set, Set Cover, and Longest Common Subsequence) that exhibit the versatility of this approach to develop important new views for computational complexity analysis.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for Dominating Set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alber, J., Fan, H., Fellows, M.R., Fernau, H., Niedermeier, R., Rosamond, F., Stege, U.: Refined search tree technique for Dominating Set on planar graphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 111–122. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: Classes of graphs with bounded treewidth. Technical Report RUU-CS-86-22, Dept. of Computer Sci., Utrecht University (1986)Google Scholar
  4. 4.
    Bodlaender, H.L.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Wareham, H.T.: The parameterized complexity of the longest common subsequence problem. Theoretical Computer Science 147, 31–54 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters 58, 171–176 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cai, L.: Parameterized complexity of Vertex Colouring. Discrete Applied Mathematics 127(1), 415–429 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, J., Kanj, I.A., Jia, W.: Vertex Cover: Further observations and further improvements. J. Algorithms 41, 280–301 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs. In: Proc. 15th SODA, pp. 830–839. SIAM, Philadelphia (2004)Google Scholar
  10. 10.
    Downey, R.G.: Parameterized complexity for the skeptic. In: Proc. 18th IEEE Annual Conference on Computational Complexity, pp. 147–169 (2003)Google Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999); 4 The latter being of particular interest when attacking W[1]-hard problemsGoogle Scholar
  12. 12.
    Ellis, J., Fan, H., Fellows, M.R.: The Dominating Set problem is fixed parameter tractable for graphs of bounded genus. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 180–189. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Fellows, M.R.: Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Fellows, M.R.: New directions and new challenges in algorithm design and complexity, parameterized. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 505–520. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Fixed-parameter algorithms for clique generation. In: CIAC 2003. LNCS, vol. 2653, pp. 108–119. Springer, Heidelberg (2003); To appear in Theory of Computing Systems.Google Scholar
  16. 16.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39(4), 321–347 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Guo, J., Niedermeier, R.: Exact algorithms and applications for Tree-like Weighted Set Cover (June 2004) (manuscript)Google Scholar
  18. 18.
    Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T., Henning, M.A.: Domination in graphs applied to electric power networks. SIAM J. Discrete Math. 15(4), 519–529 (2002)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hoffmann, M., Okamoto, Y.: The traveling salesman problem with few inner points. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 268–277. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Juedes, D., Chor, B., Fellows, M.R.: Linear kernels in linear time, or How to save k colors in O(n2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004) (to appear)CrossRefGoogle Scholar
  21. 21.
    Niedermeier, R.: Ubiquitous parameterization—invitation to fixed-parameter algorithms. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 84–103. Springer, Heidelberg (2004) (to appear)CrossRefGoogle Scholar
  22. 22.
    Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for Weighted Vertex Cover. J. Algorithms 47(2), 63–77 (2003)MATHMathSciNetGoogle Scholar
  23. 23.
    Nishimura, N., Ragde, P., Thilikos, D.M.: Fast fixed-parameter tractable algorithms for nontrivial generalizations of Vertex Cover. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 75–86. Springer, Heidelberg (2001); To appear in Discrete Applied MathematicsCrossRefGoogle Scholar
  24. 24.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet Shortest Common Supersequence and Longest Common Subsequence problems. J. Comput. Syst. Sci. 67(4), 757–771 (2003)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Robertson, N., Seymour, P.D.: Graph minors. II: Algorithmic aspects of treewidth. J. Algorithms 7, 309–322 (1986)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7, 425–440 (1986)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 548–558. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  29. 29.
    Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Telle, J.A., Proskurowski, A.: Practical algorithms on partial k-trees with an application to domination-like problems. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 610–621. Springer, Heidelberg (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jiong Guo
    • 1
  • Falk Hüffner
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany

Personalised recommendations