Perfectly Sampling k≥(8/3+o(1))Δ-Colorings in Graphs

Abstract

We present a randomized algorithm which takes as input an undirected graph G on n vertices with maximum degree ?, and a number of colors k ? (8/3 + o?(1))?, and returns - in expected time O(n?2logk) - a proper k-coloring of G distributed perfectly uniformly on the set of all proper k-colorings of G. Notably, our sampler breaks the barrier at k = 3?encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also discuss how our methods may be modified to relax the restriction on k to k ? (8/3 - ?0)?for an absolute constant ?0 > 0. As in the work of Bhandari and Chakraborty, and the pioneering work of Huber [STOC 1998], our sampler is based on Coupling from the Past [Propp&Wilson, Random Struct. Algorithms, 1995] and the bounding chain method [Huber, STOC 1998; H'aggstr'om & Nelander, Scand. J. Statist., 1999]. Our innovations include a novel bounding chain routine inspired by Jerrum's analysis of the Glauber dynamics [Random Struct. Algorithms, 1995], as well as a preconditioning routine for bounding chains which uses the algorithmic Lovász Local Lemma [Moser&Tardos, J.ACM, 2010].