## The Annals of Applied Probability

### Systematic scan for sampling colorings

#### 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

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

Digital Object Identifier
doi:10.1214/105051605000000683

Mathematical Reviews number (MathSciNet)
MR2209340

Zentralblatt MATH identifier
1095.60024

#### 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

#### References

• Aldous, D. (1982). Some inequalities for reversible Markov chains. J. London Math. Soc. (2) 25 564--576.
• Aldous, D. and Fill, J. (1996). Reversible Markov chains and random walks on graphs. Available at http://oz.berkeley.edu/users/aldous/RWG/book.html.
• Amit, Y. (1996). Convergence properties of the Gibbs sampler for perturbations of Gaussians. Ann. Statist. 24 122--140.
• Azuma, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J. 19 357--367.
• Benjamini, I., Berger, N., Hoffman, C. and Mossel, E. (2005). Mixing times of the biased card shuffling and the asymmetric exclusion process. Trans. Amer. Math. Soc. 357 3013--3029.
• Bubley, R. and Dyer, M. (1997). Path coupling: A technique for proving rapid mixing in Markov chains. Proceedings of the 38th IEEE Annual Symposium on Foundations of Computer Science 223--231.
• Cooper, C., Dyer, M. and Frieze, A. (2001). On Markov chains for randomly $H$-colouring a graph. J. Algorithms 39 117--134.
• Diaconis, P. and Ram, A. (2000). Analysis of systematic scan metropolis algorithm using Iwahori--Hecke algebra techniques. Michigan Math. J. 48 157--190.
• Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696--730.
• Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36--61.
• Durrett, R. (1991). Probability: Theory and Examples. Brooks/Cole Publishing Company.
• Dyer, M., Frieze, A. and Jerrum, M. (2002). On counting independent sets in sparse graphs. SIAM J. Comput. 31 1527--1541.
• Dyer, M., Goldberg, L. A., Greenhill, C., Jerrum, M. and Mitzenmacher, M. (2001). An extension of path coupling and its application to the Glauber dynamics for graph colorings. SIAM J. Comput. 30 1962--1975.
• Dyer, M., Goldberg, L. A., Jerrum, M. and Martin, R. (2004). Markov chain comparison. arXiv:math.PR/0410331.
• Dyer, M., Jerrum, M. and Vigoda, E. (2004). Rapidly mixing Markov chains for dismantleable constraint graphs. In Proceedings of a DIMACS/DIMATIA Workshop on Graphs, Morphisms and Statistical Physics (J. Nesetril and P. Winkler, eds.).
• Fill, J. A. (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1 62--87.
• Fishman, G. S. (1996). Coordinate selection rules for Gibbs sampling. Ann. Appl. Probab. 6 444--465.
• Glauber, R. J. (1963). Time-dependent statistics of the Ising model. J. Math. Phys. 4 294--307.
• Goldberg, L. A., Kelk, S. and Paterson, M. (2004). The complexity of choosing an $H$-colouring (nearly) uniformly at random. SIAM J. Comput. 33 416--432.
• Goldberg, L. A., Martin, R. and Paterson, M. (2004). Random sampling of $3$-colourings in $\mathbbZ^2$. Random Structures Algorithms 24 279--302.
• Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
• Jerrum, M. (1995). A very simple algorithm for estimating the number of $k$-colourings of a low-degree graph. Random Structures Algorithms 7 157--165.
• Jerrum, M. (2003). Counting, Sampling and Integrating: Algorithms and Complexity. Birkhäuser, Boston.
• Kenyon, C. and Randall, D. (2000). Personal communication.
• Luby, M., Randall, D. and Sinclair, A. J. (2001). Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31 167--192.
• Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351--370.
• Sinclair, A. and Jerrum, M. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. and Comput. 82 93--133.
• van den Berg, J. (1993). A uniqueness condition for Gibbs measures, with application to the $2$-dimensional Ising antiferromagnet. Commun. Math. Phys. 152 161--166.
• Wilson, D. B. (2004). Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274--325.