Perfect Sampling of graph k-colorings for k=o(Δ^2)

09/23/2019

∙

by   Siddharth Bhandari, et al.

∙

0

∙

share

We give an algorithm for perfect sampling from the uniform distribution on proper k-colorings of graphs of maximum degree Δ, which, for Δ∈ [17, n], terminates with a sample in expected time poly(n) time whenever k >2eΔ^2/Δ (here, n is the number of vertices in the graph). To the best of our knowledge, this is the first perfect sampling algorithm for proper k-colorings that provably terminates in expected polynomial time while requiring only k=o(Δ^2) colors in general. Inspired by the bounding chain approach pioneered independently by Häggström & Nelander (Scand. J. Statist., 1999) and Huber (STOC 1998) (who used the approach to give a perfect sampling algorithm requiring k >Δ^2 + 2Δ for its expected running time to be a polynomial), our algorithm is based on a novel bounding chain for the coloring problem.