Randomised Approximation Schemes for Tutte-Gröthendieck Invariants

  • Dominic Welsh
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.


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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

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