Theory of Computing Systems

, Volume 50, Issue 3, pp 420–432 | Cite as

A Note on Exact Algorithms for Vertex Ordering Problems on Graphs

  • Hans L. Bodlaender
  • Fedor V. Fomin
  • Arie M. C. A. Koster
  • Dieter Kratsch
  • Dimitrios M. Thilikos
Open Access


In this note, we give a proof that several vertex ordering problems can be solved in O (2 n ) time and O (2 n ) space, or in O (4 n ) time and polynomial space. The algorithms generalize algorithms for the Travelling Salesman Problem by Held and Karp (J. Soc. Ind. Appl. Math. 10:196–210, 1962) and Gurevich and Shelah (SIAM J. Comput. 16:486–502, 1987). We survey a number of vertex ordering problems to which the results apply.


Graphs Algorithms Exponential time algorithms Exact algorithms Vertex ordering problems 


  1. 1.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1–45 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L., Fomin, F.V., Koster, A.M.C.A., Kratsch, D., Thilikos, D.M.: On exact algorithms for treewidth. Technical Report UU-CS-2006-032, Department of Information and Computing Sciences, Utrecht University, Utrecht, the Netherlands (2006) Google Scholar
  3. 3.
    Bodlaender, H.L., Fomin, F.V., Koster, A.M.C.A., Kratsch, D., Thilikos, D.M.: On exact algorithms for treewidth. In: Azar, Y., Erlebach, T. (eds.) Proceedings of the 14th Annual European Symposium on Algorithms, ESA 2006. Lecture Notes in Computer Science, vol. 4168, pp. 672–683. Springer, Berlin (2006) Google Scholar
  4. 4.
    Clautiaux, F., Moukrim, A., Négre, S., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Oper. Res. 38, 13–26 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitive-search games on graphs and related parameters. Theor. Comput. Sci. 172, 233–254 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Díaz, J., Petit, J., Serna, M.: A survey of graph layout problems. ACM Comput. Surv. 34, 313–356 (2002) CrossRefGoogle Scholar
  7. 7.
    Fomin, F.V., Villanger, Y.: Treewidth computation and extremal combinatorics. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukuewics, I. (eds.) Proceedings of the 35th International Colloquium on Automata, Languages and Programming, ICALP 2008, Part I. Lecture Notes in Computer Science, vol. 5125, pp. 210–221. Springer, Berlin (2008) Google Scholar
  8. 8.
    Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: Marion, J.-Y., Schwentick, T. (eds.) Proceedings 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010. Dagstuhl Seminar Proceedings, vol. 5, pp. 383–394. Schloss Dagstuhl, Germany (2010). Leibniz-Zentrum für Informatik Google Scholar
  9. 9.
    Fomin, F.V., Kratsch, D., Todinca, I.: Exact (exponential) algorithms for treewidth and minimum fill-in. In: Díaz, J., Karhumäki, J., Lepistö, A., Sanella, D. (eds.) Proceedings of the 31st International Colloquium on Automata, Languages and Programming, ICALP 2004. Lecture Notes in Computer Science, vol. 3142, pp. 568–580. Springer, Berlin (2004) CrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Kratsch, D., Todinca, I., Villanger, Y.: Exact algorithms for treewidth and minimum fill-in. SIAM J. Comput. 38, 1058–1079 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1, 237–267 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gurevich, Y., Shelah, S.: Expected computation time for Hamiltonian path problem. SIAM J. Comput. 16, 486–502 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Held, M., Karp, R.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10, 196–210 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kinnersley, N.G.: The vertex separation number of a graph equals its path width. Inf. Process. Lett. 42, 345–350 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Koivisto, M., Parviainen, P.: A space-time tradeoff for permutation problems. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 484–492 (2010) Google Scholar
  16. 16.
    Suchan, K., Villanger, Y.: Computing pathwidth faster than 2n. In: Chen, J., Fomin, F.V. (eds.) Proceedings of the 4th International Workshop on Parameterized and Exact Computation, IWPEC 2009. Lecture Notes in Computer Science, vol. 5917, pp. 324–335. Springer, Berlin (2009) CrossRefGoogle Scholar
  17. 17.
    Villanger, Y.: Improved exponential-time algorithms for treewidth and minimum fill-in. In: Correa, J.R., Hevia, A., Kiwi, M.A. (eds.) Proceedings of the 7th Latin American Symposium on Theoretical Informatics, LATIN 2006. Lecture Notes in Computer Science, vol. 3887, pp. 800–811. Springer, Berlin (2006) CrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

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.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.Lehrstuhl II für MathematikRWTH Aachen UniversityAachenGermany
  4. 4.LITAUniversité de MetzMetz Cedex 01France
  5. 5.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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