We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for k/£~ > 1.764, the Glauber dynamics on k-colorings of a graph on n vertices with maximum degree A converges in O(nlogn) steps, assuming A = ft(log n) and that the graph is triangle-free. Previously, girth > 5 was needed. As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at thgacity ~ < (1 -~)e/A, on a regular graph G on n vertices of degree & = f~(logn) and girth >_ 6. The best known algorithm for general graphs currently assumes ,k < 2/(& -2).
For graphs of large girth, REF studies the mixing time of the Glauber dynamics and [BG08] studies deterministic correlation decay algorithms.
Coupling with the stationary distribution and improved sampling for colorings and independent sets
SODA '05
Mathematics Computer Science