 
Summary: Polynomial time randomised approximation schemes for the Tutte
polynomial of dense graphs
Noga Alon
Alan Frieze
Dominic Welsh
Abstract
The TutteGršothendieck polynomial T(G; x, y) of a graph
G encodes numerous interesting combinatorial quanti
ties 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 kcolorings, the
(all terminal) reliability probability of the graph, and var
ious other invariants the exact computation of each of
which is well known to be #Phard. Here we develop
a general technique that supplies fully polynomial ran
domised approximation schemes for approximating 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
