Degree-Based Treewidth Lower Bounds

  • Arie M. C. A. Koster
  • Thomas Wolle
  • Hans L. Bodlaender
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3503)


Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgraph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are N P-hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds.


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  1. 1.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8, 277–284 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Behzad, M., Chartrand, G., Lesniak-Foster, L.: Graphs and Digraphs. Pindle, Weber & Schmidt, Boston (1979)Google Scholar
  3. 3.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sc. 209, 1–45 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bodlaender, H.L., Koster, A.M.C.A.: On the Maximum Cardinality Search lower bound for treewidth. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 81–92. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Koster, A.M.C.A.: Safe separators for treewidth. In: Proceedings 6th Workshop on Algorithm Engineering and Experiments ALENEX2004, pp. 70–78 (2004)Google Scholar
  6. 6.
    Bodlaender, H.L., Koster, A.M.C.A., Eijkhof, F.v.d., van der Gaag, L.C.: Pre-processing for triangulation of probabilistic networks. In: Breese, J., Koller, D. (eds.) Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pp. 32–39. Morgan Kaufmann, San Francisco (2001)Google Scholar
  7. 7.
    Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 628–639. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. Technical Report UU-CS-2004-34, Dept. of Computer Science, Utrecht University, Utrecht, The Netherlands (2004)Google Scholar
  9. 9.
    Clautiaux, F., Moukrim, S.N.A., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Oper. Res. 38, 13–26 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Eijkhof, F.v.d., Bodlaender, H.L.: Safe reduction rules for weighted treewidth. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 176–185. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: proceedings UAI 2004. Uncertainty in Artificial Intelligence (2004)Google Scholar
  13. 13.
    Hicks, I.V.: Planar branch decompositions I: The ratcatcher. INFORMS Journal on Computing (2005) (to appear)Google Scholar
  14. 14.
    Koster, A.M.C.A., Bodlaender, H.L., van Hoesel, S.P.M.: Treewidth: Computational experiments. In: Broersma, H., Faigle, U., Hurink, J., Pickl, S. (eds.) Electronic Notes in Discrete Mathematics, vol. 8. Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  15. 15.
    Koster, A.M.C.A., van Hoesel, S.P.M., Kolen, A.W.J.: Solving partial constraint satisfaction problems with tree decomposition. Networks 40, 170–180 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Koster, A.M.C.A., Wolle, T., Bodlaender, H.L.: Degree-based treewidth lower bounds. Technical Report UU-CS-2004-050, Institute for Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2004)Google Scholar
  17. 17.
    Lauritzen, S.J., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. The Journal of the Royal Statistical Society. Series B (Methodological) 50, 157–224 (1988)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Lucena, B.: A new lower bound for tree-width using maximum cardinality search. SIAM J. Disc. Math. 16, 345–353 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ramachandramurthi, S.: Algorithms for VLSI Layout Based on Graph Width Metrics. PhD thesis, Computer Science Department, University of Tennessee, Knoxville, Tennessee, USA (1994)Google Scholar
  20. 20.
    Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Disc. Math. 10, 146–157 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
  24. 24.
    Wolle, T., Koster, A.M.C.A., Bodlaender, H.L.: A note on contraction degeneracy. Technical Report UU-CS-2004-042, Institute of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Arie M. C. A. Koster
    • 1
  • Thomas Wolle
    • 2
  • Hans L. Bodlaender
    • 2
  1. 1.Zuse Institute Berlin (ZIB)BerlinGermany
  2. 2.Institute of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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