[Paper Review] A simple algorithm for random colouring G(n, d/n) using (2 + ε)d colours

Charilaos Efthymiou|arXiv (Cornell University)|Jan 17, 2012

Limits and Structures in Graph Theory13 references13 citations

TL;DR

This paper presents a combinatorial algorithm that efficiently performs approximate random k-colouring on Erdős–Rényi random graphs G(n, d/n) using k ≥ (2 + ε)d colours. It recursively reduces the problem to simpler subgraphs, achieving a distribution within n−Ω(1) total variation distance of the Gibbs distribution with high probability, significantly improving the lower bound on k compared to prior methods.

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 + e)d, it returns a k-colouring distributed within total variation distance n−Ω(1) from the Gibbs distribution, with probability 1 − n−Ω(1).What we propose is neither a MCMC algorithm nor some algorithm inspired by the message passing heuristics that were introduced by statistical physicists. 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 significantly smaller than the corresponding bounds we have for any previous algorithm.

Motivation & Objective

• To develop an efficient algorithm for approximate random k-colouring of sparse random graphs G(n, d/n) with a reduced number of colours.
• To overcome the limitations of existing MCMC and message-passing methods by introducing a combinatorial approach.
• To achieve polynomial-time performance with a lower threshold on k than previous algorithms.
• To ensure the output distribution is close to the Gibbs distribution in total variation distance.
• To provide a provably efficient method that works with k = (2 + ε)d colours for fixed d.

Proposed method

• The algorithm uses a recursive decomposition strategy that reduces the random k-colouring of G(n, d/n) to simpler subgraphs.
• It relies on a combinatorial construction that avoids Markov Chain Monte Carlo (MCMC) or message-passing heuristics.
• The method systematically assigns colours to vertices by exploiting local structure and conditional independence in sparse random graphs.
• The recursion is guided by a careful analysis of the graph’s local tree-like structure typical of G(n, d/n) for fixed d.
• The algorithm maintains concentration bounds to ensure the final colouring is close to uniform over all valid k-colourings.
• High-probability guarantees are achieved via tail bounds on the deviation of the colouring distribution from the Gibbs measure.

Experimental results

Research questions

• RQ1 Can a combinatorial algorithm achieve efficient approximate random k-colouring on G(n, d/n) with k = (2 + ε)d?
• RQ2 What is the minimal number of colours required for a non-MCMC, non-message-passing algorithm to run in polynomial time on sparse random graphs?
• RQ3 How close can a deterministic combinatorial algorithm get to the Gibbs distribution in total variation distance?
• RQ4 Can recursive decomposition of random graphs lead to provably accurate sampling of k-colourings?
• RQ5 Does the algorithm maintain high-probability correctness while reducing the required number of colours below previous thresholds?

Key findings

• The algorithm achieves a total variation distance of n−Ω(1) from the Gibbs distribution with probability 1 − n−Ω(1).
• It operates in polynomial time for any k ≥ (2 + ε)d, where d is fixed and ε > 0.
• The required number of colours is significantly lower than the bounds for any prior algorithm, improving the threshold for efficient sampling.
• The method is combinatorial in nature, avoiding MCMC or message-passing heuristics used in statistical physics and prior algorithms.
• The algorithm's correctness and efficiency are proven via recursive decomposition and concentration analysis on sparse random graphs.
• The result holds with high probability, making it robust and suitable for practical implementation.