On Exact Algorithms for Treewidth

  • Hans L. Bodlaender
  • Fedor V. Fomin
  • Arie M. C. A. Koster
  • Dieter Kratsch
  • Dimitrios M. Thilikos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)


We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time O  ∗ (2 n ). This algorithm is based on the old dynamic programming method introduced by Held and Karp for the Traveling Salesman problem. We use some optimizations that do not affect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. However, our experiments show that the space used by the algorithm is an important factor to what input sizes the algorithm is effective. For this purpose, we settle the problem of computing treewidth under the restriction that the space used is only polynomial. In this direction we give a simple O  ∗ (4 n ) algorithm that requires polynomial space. We also prove that using more refined techniques with balanced separators, Treewidth can be computed in O  ∗ (2.9512 n ) time and polynomial space.


Travel Salesman Problem Exact Algorithm Dynamic Programming Algorithm Tree Decomposition Recursive Call 
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.


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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)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachoore, E.H., Bodlaender, H.L.: New upper bound heuristics for treewidth. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 217–227. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sci. 209, 1–45 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Discovering treewidth. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds.) SOFSEM 2005. LNCS, vol. 3381, pp. 1–16. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput. 31, 212–232 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bouchitté, V., Todinca, I.: Listing all potential maximal cliques of a graph. Theor. Comp. Sci. 276, 17–32 (2002)MATHCrossRefGoogle Scholar
  8. 8.
    Clautiaux, F., Moukrim, A., Négre, S., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Operations Research 38, 13–26 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitive-search games on graphs and related parameters. Theor. Comp. Sci. 172, 233–254 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. Bulletin of the EATCS 87, 47–77 (2005)MATHMathSciNetGoogle Scholar
  11. 11.
    Fomin, F.V., Kratsch, D., Todinca, I.: Exact (exponential) algorithms for treewidth and minimum fill-in. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 568–580. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence UAI 2004, Arlington, Virginia, USA, pp. 201–208. AUAI Press (2004)Google Scholar
  13. 13.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)MATHGoogle Scholar
  14. 14.
    Gurevich, Y., Shelah, S.: Expected computation time for Hamiltonian path problem. SIAM J. Comput. 16, 486–502 (1987)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Held, M., Karp, R.: A dynamic programming approach to sequencing problems. J. SIAM 10, 196–210 (1962)MATHMathSciNetGoogle Scholar
  16. 16.
    Koster, A.M.C.A., Wolle, T., Bodlaender, H.L.: Degree-based treewidth lower bounds. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 101–112. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Shoikhet, K., Geiger, D.: A practical algorithm for finding optimal triangulations. In: Proc. National Conference on Artificial Intelligence (AAAI 1997), pp. 185–190. Morgan Kaufmann, San Francisco (1997)Google Scholar
  19. 19.
  20. 20.
    Villanger, Y.: Improved exponential-time algorithms for treewidth and minimum fill-in. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 800–811. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  • Fedor V. Fomin
    • 2
  • Arie M. C. A. Koster
    • 3
  • Dieter Kratsch
    • 4
  • Dimitrios M. Thilikos
    • 5
  1. 1.Institute of Information and Computing SciencesUtrecht UniversityTB UtrechtThe Netherlands
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlinGermany
  4. 4.LITA, Université Paul Verlaine – MetzMetzFrance
  5. 5.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de Catalunya, Edifici ΩBarcelonaSpain

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