Switching Colouring of G(n, d/n) for Sampling up to Gibbs Uniqueness Threshold.

Approximate random k-colouring of a graph G = (V, E), efficiently, is a very well studied problem in computer science and statistical physics. It amounts to constructing, in polynomial time, a k-colouring of G which is distributed close to Gibbs distribution. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n, d/n), where d is fixed. This paper improves on the approximate sampling colouring algorithm proposed in SODA 2012. We provide improved performance guarantees for this efficient algorithm, as we reduce the lower bound of the number of colours required by a factor of 1/2. In particular, we show the following statement for the accuracy of algorithm: For typical instances of G(n, d/n) the algorithm outputs a k-colouring of G(n, d/n) which is asymptotically uniform as long k >= (1 + epsilon)d. For the improvement we make an extensive use of the spatial correlation decay properties of the Gibbs distribution and the local treelike structure of the underlying graph.