Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

TL;DR

This paper introduces a novel, efficient pattern matching algorithm to compute chromatic polynomials of graphs, significantly outperforming existing methods and enabling the analysis of complex three-dimensional lattice structures.

Contribution

The paper presents a vertex-oriented symbolic pattern matching algorithm that efficiently computes chromatic polynomials, outperforming traditional methods and enabling analysis of larger, more complex graphs.

Findings

Outperforms standard methods by several orders of magnitude

Successfully computes chromatic polynomials of 3D cubic lattices

Enables analysis of complex disordered systems in physics and chemistry

Abstract

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.