Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization

  • Hans L. Bodlaender
  • Bart M. P. Jansen
  • Stefan Kratsch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


Using the framework of kernelization we study whether efficient preprocessing schemes for the Treewidth problem can give provable bounds on the size of the processed instances. Assuming the AND-distillation conjecture to hold, the standard parameterization of Treewidth does not have a kernel of polynomial size and thus instances (G,k) of the decision problem of Treewidth cannot be efficiently reduced to equivalent instances of size polynomial in k. In this paper, we consider different parameterizations of Treewidth. We show that Treewidth has a kernel with \(\mathcal{O}(\ell^3)\) vertices, where ℓ denotes the size of a vertex cover, and a kernel with \(\mathcal{O}(\ell^4)\) vertices, where ℓ denotes the size of a feedback vertex set. This implies that given an instance (G,k) of Treewidth we can efficiently reduce its size to \(\mathcal{O}((\ell^*)^4)\) vertices, where ℓ* is the size of a minimum feedback vertex set in G. In contrast, we show that Treewidth parameterized by the vertex-deletion distance to a co-cluster graph and Weighted Treewidth parameterized by the size of a vertex cover do not have polynomial kernels unless NP ⊆ coNP/poly. Treewidth parameterized by the target value plus the deletion distance to a cluster graph has no polynomial kernel unless the AND-distillation conjecture does not hold.


Vertex Cover Polynomial Kernel Tree Decomposition Common Neighbor Chordal Graph 
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.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic and Discrete Methods 8, 277–284 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Becker, A., Geiger, D.: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence 83, 167–188 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Necessary edges in k-chordalizations of graphs. Journal of Combinatorial Optimization 7, 283–290 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. Journal of Computer and System Sciences 75, 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: STACS 2011, pp. 165–176 (2011)Google Scholar
  7. 7.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Preprocessing for treewidth: A combinatorial analysis through kernelization. CoRR, abs/1104.4217 (2011)Google Scholar
  8. 8.
    Bodlaender, H.L., Koster, A.M.C.A.: Safe separators for treewidth. Discrete Mathematics 306, 337–350 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bodlaender, H.L., Koster, A.M.C.A., van den Eijkhof, F.: Pre-processing rules for triangulation of probabilistic networks. Computational Intelligence 21(3), 286–305 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. Journal of Graph Algorithms and Applications 10, 5–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM Journal on Discrete Mathematics 6, 181–188 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bodlaender, H.L., Rotics, U.: Computing the treewidth and the minimum fill-in with the modular decomposition. Algorithmica 36, 375–408 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Clautiaux, F., Carlier, J., Moukrim, A., Négre, S.: New lower and upper bounds for graph treewidth. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 70–80. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Clautiaux, F., Moukrim, A., Négre, S., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Operations Research 38, 13–26 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    van den Eijkhof, F., Bodlaender, H.L., Koster, A.M.C.A.: Safe reduction rules for weighted treewidth. Algorithmica 47, 138–158 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007)CrossRefGoogle Scholar
  17. 17.
    Lagergren, J., Arnborg, S.: Finding minimal forbidden minors using a finite congruence. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 532–543. Springer, Heidelberg (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  • Bart M. P. Jansen
    • 1
  • Stefan Kratsch
    • 1
  1. 1.Utrecht UniversityThe Netherlands

Personalised recommendations