Discrete Probability and Algorithms pp 133-148 | Cite as

# Randomised Approximation Schemes for Tutte-Gröthendieck Invariants

## Abstract

*G.*

- (i)
What is the number of connected subgraphs of

*G*? - (ii)
How many subgraphs of

*G*are forests? - (iii)
How many acyclic orientations has

*G?*

*x*,

*y*)

*-*plane — in other words is a Tutte-Gröthendieck invariant. Other invariants include:

- (iv)
the chromatic and flow polynomials of a graph;

- (v)
the partition function of a

*Q*-state Potts model; - (vi)
the Jones polynomial of an alternating link;

- (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.

## Keywords

Partition Function Planar Graph Linear Code Connected Subgraph Dense Graph## Preview

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