A simple algorithm for random colouring G(n, d/n) using (2+\epsilon)d colours

TL;DR

This paper introduces a simple, combinatorial algorithm for approximately uniformly sampling k-colourings of Erdős–Rényi random graphs G(n,d/n) using slightly more than 2d colours, with strong theoretical guarantees.

Contribution

It presents a novel, efficient combinatorial algorithm for approximate random k-colouring of G(n,d/n), improving bounds on the number of colours needed compared to prior methods.

Findings

Algorithm achieves total variation distance n^{-Omega(1)} from Gibbs distribution.

Works for k > (2 + epsilon)d with high probability.

Lower bounds on colour count are significantly improved.

Abstract

Approximate random k-colouring of a graph G=(V,E) is a very well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution, i.e. the uniform distribution over all the k-colourings of G. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,p), where p=d/n and d is fixed. We propose a novel efficient algorithm for approximate random k-colouring with the following properties: given an instance of G(n,d/n) and for any k>(2+\epsilon)d, it returns a k-colouring distributed within total variation distance n^{-Omega(1)} from the Gibbs distribution, with probability 1-n^{-Omega(1)}. What we propose is neither a MCMC algorithm nor some algorithm inspired by the message passing heuristics that were introduced by statistical physicist. Our algorithm is of combinatorial nature. It is based on a rather simple recursion which reduces the random k-colouring of G(n,d/n) to random k-colouring simpler subgraphs first. The lower bound on the number of colours for our algorithm to run in polynomial time is dramatically smaller than the corresponding bounds we have for any previous algorithm.