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

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 $k \geq (8/3 + o_Δ(1))Δ$, and returns -- in expected time $\tilde{O}(nΔ^{2}\log{k})$ -- 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 $k \geq (8/3 - ε_0)Δ$ 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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