Deterministic counting of graph colourings using sequences of subgraphs

Abstract

In this paper we propose a deterministic algorithm for approximately counting the \(k\)-colourings of sparse random graphs \(G(n,d/n)\). In particular, our algorithm computes in polynomial time a \((1\pm n^{-\Omega(1)})\)approximation of the logarithm of the number of \(k\)-colourings of \(G(n,d/n)\) for \(k\geq (2+\epsilon) d\) with high probability over the graph instances. Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA '06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation decay} to compute {\em deterministically} marginals of {\em Gibbs distribution}. We develop a scheme whose accuracy depends on {\em non-reconstruction} of the colourings of \(G(n,d/n)\), rather than {\em uniqueness} that are required in previous works. This leaves open the possibility for our schema to be sufficiently accurate even for \(k<d\). The set up for establishing correlation decay is as follows: Given \(G(n,d/n)\), we alter the graph structure in some specific region \(\Lambda\) of the graph by deleting edges between vertices of \(\Lambda\). Then we show that the effect of this change on the marginals of Gibbs distribution, diminishes as we move away from \(\Lambda\). Our approach is novel and suggests a new context for the study of deterministic counting algorithms.