Approximating the number of acyclic orientations for a class of sparse graphs

Bordewich, M.

Abstract

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 \cite{vertiganandwelsh} 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 randomised 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 $(2,0)$.

Citation

Bordewich, M. (2004). Approximating the number of acyclic orientations for a class of sparse graphs. Combinatorics, Probability and Computing, 13(1), 1-16. https://doi.org/10.1017/s0963548303005844

• Journal Article Type: Article
• Publication Date: 2004-01
• Deposit Date: Jun 18, 2008
• Publicly Available Date: Jun 18, 2008
• Journal: Combinatorics, Probability and Computing
• Print ISSN: 0963-5483
• Electronic ISSN: 1469-2163
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 13
• Issue: 1
• Pages: 1-16
• DOI: https://doi.org/10.1017/s0963548303005844
• Keywords: Tutte polynomial, FPRAS, Approximation, Complexity.
• Public URL: https://durham-repository.worktribe.com/output/1590287