Random sampling of colourings of sparse random graphs with a constant number of colours

TL;DR

This paper introduces a simple, efficient algorithm for nearly uniformly sampling proper k-colourings of sparse random graphs, using novel spatial mixing properties, and improves on previous MCMC-based methods by requiring a constant number of colours.

Contribution

It presents a new algorithm for sampling colourings that does not rely on Markov Chain Monte Carlo, with a novel proof based on spatial mixing properties, applicable to a constant number of colours.

Findings

Algorithm works with high probability for sparse random graphs.

Provides a novel proof of correctness based on spatial mixing.

Improves previous methods by requiring only a constant number of colours.

Abstract

In this work we present a simple and efficient algorithm which, with high probability, provides an almost uniform sample from the set of proper k-colourings on an instance of a sparse random graph G(n,d/n), where k=k(d) is a sufficiently large constant. Our algorithm is not based on the Markov Chain Monte Carlo method (M.C.M.C.). Instead, we provide a novel proof of correctness of our Algorithm that is based on interesting "spatial mixing" properties of colourings of G(n,d/n). Our result improves upon previous results (based on M.C.M.C.) that required a number of colours growing unboundedly with n.