Randomised Approximation Schemes for Tutte-Gröthendieck Invariants

  • Dominic Welsh
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 72)


Consider the following very simple counting problems associated with a graph G.
  1. (i)

    What is the number of connected subgraphs of G?

  2. (ii)

    How many subgraphs of G are forests?

  3. (iii)

    How many acyclic orientations has G?

Each of these is a special case of the general problem of evaluating the Tutte polynomial of a graph (or matroid) at a particular point of the (x, y)-plane — in other words is a Tutte-Gröthendieck invariant. Other invariants include:
  1. (iv)

    the chromatic and flow polynomials of a graph;

  2. (v)

    the partition function of a Q-state Potts model;

  3. (vi)

    the Jones polynomial of an alternating link;

  4. (vii)

    the weight enumerator of a linear code over GF(q).


It has been shown that apart from a few special points and 2 special hyperbolae, the exact evaluation of any such invariant is #P-hard even for the very restricted class of planar bipartite graphs. However the question of which points have a fully polynomial randomised approximation scheme is wide open. I shall discuss this problem and give a survey of what is currently known.


Partition Function Planar Graph Linear Code Connected Subgraph Dense 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.
    Annan, J.D., A randomised approximation algorithm for counting the number of forests in dense graphs, Combinatorics, Probability and Computer Science, (submitted) (1993).Google Scholar
  2. 2.
    Brylawski, T.H. And Oxley, J.G., The Tutte polynomial and its applications, Matroid Applications (ed. N. White), Cambridge Univ. Press (1992), pp. 123–225.Google Scholar
  3. 3.
    Colbourn, C.J., The Combinatorics of Network Reliability, Oxford University Press (1987).Google Scholar
  4. 4.
    Edwards, K., The complexity of colouring problems on dense graphs, Theoretical Computer Science, 43 (1986), pp. 337–343.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Feder, T. and Mihail, M., Balanced matroids, Proceedings of 24th Annual ACM Symposium on the Theory of Computing (1992), pp. 26–38.Google Scholar
  6. 6.
    Garey, M.R. and Johnson, D.S., Computers and Intractability-A guide to the theory of NP-completeness. Freeman, San Francisco (1979).Google Scholar
  7. 7.
    Grötschel M., Lovász, L. and Schrijver, A., Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin (1988).CrossRefMATHGoogle Scholar
  8. 8.
    Jaeger, F., Nowhere zero flow problems, Selected Topics in Graph Theory 3, (ed. L. Beineke and R.J. Wilson) Academic Press, London (1988), pp. 71–92.Google Scholar
  9. 9.
    Jaeger F., Vertigan, D.L., and Welsh, D.J.A., On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Camb. Phil. Soc. 108 (1990), pp. 35–53.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jerrum, M.R. and Sinclair, A., Polynomial-time approximation algorithms for the Ising model, Proc. 17th Icalp, Eatcs (1990), pp. 462–475.Google Scholar
  11. 11.
    Jerrum, M.R., Valiant, L.G. and Vazirani, V.V., Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science, 43 (1990), pp. 169–188.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mihail, M. and Winkler, P., On the number of Eulerian orientations of a graph, Bellcore Technical Memorandum TM-ARH-018829 (1991).Google Scholar
  13. 13.
    Moore, E.F. and Shannon, C.E., Reliable circuits using less reliable components, Journ. Franklin Instit., 262 (1956), pp. 191–208, 281-297.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Oxley, J.G., Matroid Theory, Oxford Univ. Press (1992).Google Scholar
  15. 15.
    Oxley, J.G. and Welsh, D.J.A., (1979) The Tutte polynomial and percolation, Graph Theory and Related Topics (eds. J.A. Bondy and U.S.R. Murty), Academic Press, London (1979), pp. 329–339.Google Scholar
  16. 16.
    Rosenstiehl, P. and Read, R.C., On the principal edge tripartition of a graph, Ann. Discrete Math., 3 (1978), pp. 195–226.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Seymour, P.D., Nowhere-zero 6-flows, J. Comb. Theory B, 30, (1981), pp. 130–135.MathSciNetGoogle Scholar
  18. 18.
    Seymour, P.D. and Welsh, D.J.A., Combinatorial applications of an inequality from statistical mechanics, Math. Proc. Camb. Phil. Soc, 77 (1975), pp. 485–497.Google Scholar
  19. 19.
    Spencer, J. (private communication) (1993).Google Scholar
  20. 20.
    Thistlethwaite, M.B., A spanning tree expansion of the Jones polynomial, Topology, 26 (1987), pp. 297–309.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Vertigan, D.L. and Welsh, D.J.A., The computational complexity of the Tutte plane: the bipartite case, Probability, Combinatorics and Computer Science, 1 (1992), pp. 181–187.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Welsh, D.J.A., Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series 186, Cambridge University Press (1993).Google Scholar
  23. 23.
    Welsh, D.J.A., Randomised approximation in the Tutte plane, Combinatorics, Probability and Computing, 3 (1994), pp. 137–143.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zaslavsky, T., Facing up to arrangements: face count formulas for partitions of spaces by hyperplanes, Memoirs of American Mathematical Society, 154 (1975).Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Dominic Welsh
    • 1
  1. 1.Merton College and the Mathematical Institute, Oxford UniversityOxfordEngland

Personalised recommendations