Open Access
August 2006 Coupling with the stationary distribution and improved sampling for colorings and independent sets
Thomas P. Hayes, Eric Vigoda
Ann. Appl. Probab. 16(3): 1297-1318 (August 2006). DOI: 10.1214/105051606000000330


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).


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Thomas P. Hayes. Eric Vigoda. "Coupling with the stationary distribution and improved sampling for colorings and independent sets." Ann. Appl. Probab. 16 (3) 1297 - 1318, August 2006.


Published: August 2006
First available in Project Euclid: 2 October 2006

zbMATH: 1120.60067
MathSciNet: MR2260064
Digital Object Identifier: 10.1214/105051606000000330

Primary: 60J10
Secondary: 68W20

Keywords: Coupling method , hard-core model , mixing time of Markov chains , random colorings

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.16 • No. 3 • August 2006
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