Polynomial time randomised approximation schemes for the Tutte polynomial of dense graphs

The Tutte-Grothendieck polynomial T(G; x, y) of a graph G encodes numerous interesting combinatorial 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 approximating the value of T(G; x, y) for any dense graph G, that is, any graph on n vertices whose minimum degree is Ω(n), whenever x 1 and y 1, and in various additional points. 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.