Perfectly Sampling k≥ (8/3 +o(1))Δ-Colorings in Graphs
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We present a randomized algorithm which takes as input an undirected graph G on n vertices with maximum degree Δ, and a number of colors k ≥ (8/3 + o_Δ(1))Δ, and returns – in expected time Õ(nΔ^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 = 3Δ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also sketch how to modify our methods to relax the restriction on k to k ≥ (8/3 - ϵ_0)Δ for an absolute constant ϵ_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; Häggström 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].
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