The Annals of Applied Probability

Systematic scan for sampling colorings

Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum

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We address the problem of sampling colorings of a graph G by Markov chain simulation. For most of the article we restrict attention to proper q-colorings of a path on n vertices (in statistical physics terms, the one-dimensional q-state Potts model at zero temperature), though in later sections we widen our scope to general “H-colorings” of arbitrary graphs G. Existing theoretical analyses of the mixing time of such simulations relate mainly to a dynamics in which a random vertex is selected for updating at each step. However, experimental work is often carried out using systematic strategies that cycle through coordinates in a deterministic manner, a dynamics sometimes known as systematic scan. The mixing time of systematic scan seems more difficult to analyze than that of random updates, and little is currently known. In this article we go some way toward correcting this imbalance. By adapting a variety of techniques, we derive upper and lower bounds (often tight) on the mixing time of systematic scan. An unusual feature of systematic scan as far as the analysis is concerned is that it fails to be time reversible.

Article information

Ann. Appl. Probab., Volume 16, Number 1 (2006), 185-230.

First available in Project Euclid: 6 March 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05C15: Coloring of graphs and hypergraphs 60C15 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 68W20: Randomized algorithms 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Glauber dynamics graph homomorphisms mixing time Potts model spin systems systematic scan


Dyer, Martin; Goldberg, Leslie Ann; Jerrum, Mark. Systematic scan for sampling colorings. Ann. Appl. Probab. 16 (2006), no. 1, 185--230. doi:10.1214/105051605000000683.

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