TR19-164 Authors: Siddharth Bhandari, Sayantan Chakraborty

Publication: 17th November 2019 12:56

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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 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 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 for $\Delta = \omega(1)$ and for any $k \leq \Delta^2+2\Delta$.

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 H\"aggstr\"om \& Nelander~(Scand.{} J.{} Statist., 1999) and Huber~(STOC 1998), our algorithm is based on a novel bounding chain for the coloring problem.