Summary: Polynomial time randomised approximation schemes for
Tutte-Gršothendieck invariants: the dense case
The Tutte-Gršothendieck polynomial T(G; x, y) of a graph G encodes numerous interesting combi-
natorial quantities associated with the graph. Its evaluation in various points in the (x, y) plane give
the number of spanning forests of the graph, the number of its strongly connected orientations, the
number of its proper k-colorings, the (all terminal) reliability probability of the graph, and various
other invariants the exact computation of each of which is well known to be #P-hard. Here we develop
a general technique that supplies fully polynomial randomised approximation schemes for approximat-
ing the value of T(G; x, y) for any dense graph G, that is, any graph on n vertices whose minimum
degree is (n), whenever x 1 and y > 1, and in various additional points. Annan  has dealt with
the case y = 1, x 1. This region includes evaluations of reliability and partition functions of the
ferromagnetic Q-state Potts model. Extensions to linear matroids where T specialises to the weight
enumerator of linear codes are considered as well.
Consider the following very simple counting problems associated with a graph G.
(i) What is the number of connected subgraphs of G?