Improved bounds for perfect sampling of k-colorings in graphs

Abstract

We present a randomized algorithm that takes as input an undirected n-vertex graph G with maximum degree Δ and an integer k > 3Δ, and returns a random proper k-coloring of G. The distribution of the coloring is perfectly uniform over the set of all proper k-colorings; the expected running time of the algorithm is poly(k,n)=O(nΔ2· log(k)). This improves upon a result of Huber (STOC 1998) who obtained a polynomial time perfect sampling algorithm for k>Δ2+2Δ. Prior to our work, no algorithm with expected running time poly(k,n) was known to guarantee perfectly sampling with sub-quadratic number of colors in general. Our algorithm (like several other perfect sampling algorithms including Huber’s) is based on the Coupling from the Past method. Inspired by the bounding chain approach, pioneered independently by Huber (STOC 1998) and H'aggstr'om & Nelander (Scand. J. Statist., 1999), we employ a novel bounding chain to derive our result for the graph coloring problem.