Computing the Tutte Polynomial in Vertex-Exponential Time

The deletion–contraction algorithm is perhapsthe 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.

Authors

• Andreas Björklund
• Thore Husfeldt
• Petteri Kaski
• Mikko Koivisto

Keywords

• Polynomials
• Computer science
• Partitioning algorithms
• Physics computing
• Graph theory
• Tree graphs
• Quantum computing
• Approximation algorithms
• Reliability theory
• Information technology
• Tutte Polynomial
• Statistical Physics
• Graph Properties
• Space Of Polynomials
• Time And Space
• Running Time
• Potential Model
• Ising Model
• Exact Algorithm
• Breadth-first Search
• Algorithm Running
• Hyperbola
• Polynomial Ring
• Constraint Satisfaction Problem
• Components Of The Graph
• Polynomial Interpolation
• Planar Graphs
• Induced Subgraph
• Class Of Graphs
• Subset Of Vertices
• Ring Elements
• Graph Parameters
• Theoretical Computer Science
• Exact algorithms
• exponential-time algorithms
• Potts model