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 $\Delta$ and an integer $k > 3\Delta$, and returns a random proper $k$coloring of $G$. The distribution of the coloring is \emph{perfectly} uniform over the set of all proper $k$colorings; the expected running time of the algorithm is $\mathrm{poly}(k,n)=\widetilde{O}(n\Delta^2\cdot \log(k))$. This improves upon a result of Huber~(STOC 1998) who obtained a polynomial time perfect sampling algorithm for $k>\Delta^2+2\Delta$. Prior to our work, no algorithm with expected running time $\mathrm{poly}(k,n)$ was known to guarantee perfectly sampling with subquadratic 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 \emph{bounding chain} approach, pioneered independently by Huber~(STOC 1998) and Häggström \& Nelander~(Scand.{} J.{} Statist., 1999), we employ a novel bounding chain to derive our result for the graph coloring problem.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.10323
 Bibcode:
 2019arXiv190910323B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics
 EPrint:
 Rewrite of previous version