Algorithm for computing the partition function of the Potts model for SP-graphs

TL;DR

This paper introduces a linear-time algorithm for exactly computing the Potts model partition function on series-parallel graphs, leveraging graph simplification techniques to improve computational efficiency.

Contribution

The paper presents the first linear-time algorithm for calculating the Potts model partition function specifically on SP-graphs, using edge replacement methods.

Findings

Algorithm runs in linear time for SP-graphs.

Efficient computation of the Potts partition function on series-parallel graphs.

Graph simplification reduces computational complexity.

Abstract

The q-state Potts model is a fundamental framework in statistical physics and graph theory, with its partition function encoding rich information about spin configurations. The multivariate Tutte polynomial (known as the partition function of the Potts model) can be defined on an arbitrary finite graph $G$ and encodes a lot of important combinatorial information about the graph. As a special case, it contains the familiar Tutte polynomial with two variables and, consequently, its specialization with one variable, such as the chromatic polynomial, the flow polynomial and the reliability polynomial. The main goal of this paper is to present an efficient algorithm for computing the Potts model partition function on SP-graphs (series-parallel graphs) with arbitrary weights. The algorithm for SP-graphs is based on simplifying the graph by replacing several edges with a single edge of equivalent weight, which significantly reduces computational complexity. In this paper, we present a linear-time algorithm for exactly computing the Potts model partition function on series-parallel graphs (SP-graphs).