A simple algorithm for sampling colourings of $G(n,d/n)$ up to Gibbs Uniqueness Threshold

TL;DR

This paper introduces a novel, efficient algorithm for approximately sampling proper $k$-colourings of Erd ext{"o}s-Rényi random graphs near the Gibbs uniqueness threshold, bypassing traditional MCMC and message passing methods.

Contribution

The authors develop a new algorithm that efficiently samples from the Gibbs distribution for $k$-colourings of $G(n,d/n)$ graphs when $k \\geq (1+\\epsilon)d$, using a process of edge removal and incremental re-addition.

Findings

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

Works with high probability over random graph instances.

Operates efficiently without MCMC or message passing techniques.

Abstract

Approximate random $k$-colouring of a graph $G$ is a well studied problem in computer science and statistical physics. It amounts to constructing a $k$-colouring of $G$ which is distributed close to {\em Gibbs distribution} in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erd\H{o}s-R\'enyi random graph $G(n,d/n)$, where $d$ is a sufficiently large constant. We propose a novel efficient algorithm for approximate random $k$-colouring $G(n,d/n)$ for any $k\geq (1+\epsilon)d$. To be more specific, with probability at least $1-n^{-\Omega(1)}$ over the input instances $G(n,d/n)$ and for $k\geq (1+\epsilon)d$, the algorithm returns a $k$-colouring which is distributed within total variation distance $n^{-\Omega(1)}$ from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a "simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution.