Computing the Tutte polynomial in vertex-exponential time

TL;DR

This paper introduces a significantly faster algorithm for computing the Tutte polynomial of a graph, which also applies to related invariants, outperforming traditional deletion--contraction methods both theoretically and practically.

Contribution

The authors develop a new algorithm that computes the Tutte polynomial in time proportional to the number of connected vertex sets, improving over the deletion--contraction method.

Findings

Algorithm runs in polynomial factor of connected vertex sets

Outperforms deletion--contraction in practice

Provides polynomial-space variant and extends to cover polynomial

Abstract

The deletion--contraction algorithm is perhaps the most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin--Kasteleyn in statistical physics. Prior to this work, deletion--contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph. Here, we give a substantially faster algorithm that computes the Tutte polynomial--and hence, all the aforementioned invariants and more--of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Graham's cover polynomial. An implementation of the algorithm outperforms deletion--contraction also in practice.