The Annals of Applied Probability

Coupling with the stationary distribution and improved sampling for colorings and independent sets

Thomas P. Hayes and Eric Vigoda

Full-text: Open access

Abstract

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).

Article information

Source
Ann. Appl. Probab. Volume 16, Number 3 (2006), 1297-1318.

Dates
First available in Project Euclid: 2 October 2006

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1159804982

Digital Object Identifier
doi:10.1214/105051606000000330

Mathematical Reviews number (MathSciNet)
MR2260064

Zentralblatt MATH identifier
1120.60067

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68W20: Randomized algorithms

Keywords
Mixing time of Markov chains coupling method random colorings hard-core model

Citation

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.


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