The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 16, Number 3 (2006), 1297-1318.
Coupling with the stationary distribution and improved sampling for colorings and independent sets
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/Δ>1.764, the Glauber dynamics on k-colorings of a graph on n vertices with maximum degree Δ converges in O(nlog n) steps, assuming Δ=Ω(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 fugacity λ<(1−ɛ)e/Δ, on a regular graph G on n vertices of degree Δ=Ω(log n) and girth ≥6. The best known algorithm for general graphs currently assumes λ<2/(Δ−2).
Ann. Appl. Probab. Volume 16, Number 3 (2006), 1297-1318.
First available in Project Euclid: 2 October 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68W20: Randomized algorithms
Hayes, Thomas P.; Vigoda, Eric. Coupling with the stationary distribution and improved sampling for colorings and independent sets. Ann. Appl. Probab. 16 (2006), no. 3, 1297--1318. doi:10.1214/105051606000000330. http://projecteuclid.org/euclid.aoap/1159804982.