Approximating the Number of Acyclic Orientations for a Class of Sparse Graphs
The Tutte polynomial $T(G;x,y)$ of a graph evaluates to many interesting combinatorial quantities at various points in the $(x,y)$ plane, including the number of spanning trees, number of forests, number of acyclic orientations, the reliability polynomial, the partition function of the Q-state Potts model of a graph, and the Jones polynomial of an alternating link. The exact computation of $T(G;x,y)$ has been shown by Vertigan and Welsh  to be #P-hard at all but a few special points and on two hyperbolae, even in the restricted class of planar bipartite graphs. Attention has therefore been focused on approximation schemes. To date, positive results have been restricted to the upper half plane $y>1$, and most results have relied on a condition of sufficient denseness in the graph. In this paper we present an approach that yields a fully polynomial randomized approximation scheme for $T(G;x,y)$ for $x>1,\ y=1$, and for $T(G;2,0)$, in a class of sparse graphs. This is the first positive result that includes the important point $(2,0)$.(Received May 14 2002)
(Revised March 31 2003)
1 Research funded by the EPSRC and Vodafone, and supported in part by ESPRIT Project RAND-APX.