The Annals of Applied Probability

Systematic scan for sampling colorings

Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum

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Abstract

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

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

Dates
First available in Project Euclid: 6 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1141654285

Digital Object Identifier
doi:10.1214/105051605000000683

Mathematical Reviews number (MathSciNet)
MR2209340

Zentralblatt MATH identifier
1095.60024

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1141654285


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