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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References

  • Aldous, D. (1982). Some inequalities for reversible Markov chains. J. London Math. Soc. (2) 25 564–576.
    Mathematical Reviews (MathSciNet): MR657512
    Zentralblatt MATH: 0489.60077
    Digital Object Identifier: doi:10.1112/jlms/s2-25.3.564
  • 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.
    URL: Link to item
  • Amit, Y. (1996). Convergence properties of the Gibbs sampler for perturbations of Gaussians. Ann. Statist. 24 122–140.
    Mathematical Reviews (MathSciNet): MR1389883
    Zentralblatt MATH: 0854.60066
    Digital Object Identifier: doi:10.1214/aos/1033066202
    Project Euclid: euclid.aos/1033066202
  • Azuma, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J. 19 357–367.
    Mathematical Reviews (MathSciNet): MR221571
    Zentralblatt MATH: 0178.21103
    Digital Object Identifier: doi:10.2748/tmj/1178243286
    Project Euclid: euclid.tmj/1178243286
  • 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.
    Mathematical Reviews (MathSciNet): MR2135733
    Zentralblatt MATH: 1071.60095
    Digital Object Identifier: doi:10.1090/S0002-9947-05-03610-X
  • 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.
    Zentralblatt MATH: 1321.68378
  • Cooper, C., Dyer, M. and Frieze, A. (2001). On Markov chains for randomly $H$-colouring a graph. J. Algorithms 39 117–134.
    Mathematical Reviews (MathSciNet): MR1823658
    Digital Object Identifier: doi:10.1006/jagm.2000.1142
  • Diaconis, P. and Ram, A. (2000). Analysis of systematic scan metropolis algorithm using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 157–190.
    Mathematical Reviews (MathSciNet): MR1786485
    Zentralblatt MATH: 0998.60069
    Digital Object Identifier: doi:10.1307/mmj/1030132713
    Project Euclid: euclid.mmj/1030132713
  • Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
    Mathematical Reviews (MathSciNet): MR1233621
    Zentralblatt MATH: 0799.60058
    Digital Object Identifier: doi:10.1214/aoap/1177005359
    Project Euclid: euclid.aoap/1177005359
  • Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36–61.
    Mathematical Reviews (MathSciNet): MR1097463
    Zentralblatt MATH: 0731.60061
    Digital Object Identifier: doi:10.1214/aoap/1177005980
    Project Euclid: euclid.aoap/1177005980
  • Durrett, R. (1991). Probability: Theory and Examples. Brooks/Cole Publishing Company.
    Mathematical Reviews (MathSciNet): MR1068527
    Zentralblatt MATH: 0709.60002
  • Dyer, M., Frieze, A. and Jerrum, M. (2002). On counting independent sets in sparse graphs. SIAM J. Comput. 31 1527–1541.
    Mathematical Reviews (MathSciNet): MR1936657
    Zentralblatt MATH: 1041.68045
    Digital Object Identifier: doi:10.1137/S0097539701383844
  • 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.
    Mathematical Reviews (MathSciNet): MR1856564
    Zentralblatt MATH: 0999.05035
    Digital Object Identifier: doi:10.1137/S0097539700372708
  • 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.).
    Zentralblatt MATH: 1061.05031
  • 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.
    Mathematical Reviews (MathSciNet): MR1097464
    Zentralblatt MATH: 0726.60069
    Digital Object Identifier: doi:10.1214/aoap/1177005981
    Project Euclid: euclid.aoap/1177005981
  • Fishman, G. S. (1996). Coordinate selection rules for Gibbs sampling. Ann. Appl. Probab. 6 444–465.
    Mathematical Reviews (MathSciNet): MR1398053
    Zentralblatt MATH: 0855.60060
    Digital Object Identifier: doi:10.1214/aoap/1034968139
    Project Euclid: euclid.aoap/1034968139
  • Glauber, R. J. (1963). Time-dependent statistics of the Ising model. J. Math. Phys. 4 294–307.
    Mathematical Reviews (MathSciNet): MR148410
    Zentralblatt MATH: 0145.24003
    Digital Object Identifier: doi:10.1063/1.1703954
  • 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.
    Mathematical Reviews (MathSciNet): MR2048449
    Digital Object Identifier: doi:10.1137/S0097539702408363
  • Goldberg, L. A., Martin, R. and Paterson, M. (2004). Random sampling of $3$-colourings in $\mathbb{Z}^2$. Random Structures Algorithms 24 279–302.
    Mathematical Reviews (MathSciNet): MR2068870
    Digital Object Identifier: doi:10.1002/rsa.20002
  • Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1782847
  • 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.
    Mathematical Reviews (MathSciNet): MR1369061
    Digital Object Identifier: doi:10.1002/rsa.3240070205
  • Jerrum, M. (2003). Counting, Sampling and Integrating: Algorithms and Complexity. Birkhäuser, Boston.
    Mathematical Reviews (MathSciNet): MR1960003
  • 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.
    Mathematical Reviews (MathSciNet): MR1857394
    Zentralblatt MATH: 0992.82013
    Digital Object Identifier: doi:10.1137/S0097539799360355
  • Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351–370.
    Mathematical Reviews (MathSciNet): MR1211324
    Zentralblatt MATH: 0801.90039
    Digital Object Identifier: doi:10.1017/S0963548300000390
  • Sinclair, A. and Jerrum, M. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. and Comput. 82 93–133.
    Mathematical Reviews (MathSciNet): MR1003059
    Zentralblatt MATH: 0668.05060
    Digital Object Identifier: doi:10.1016/0890-5401(89)90067-9
  • 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.
    Mathematical Reviews (MathSciNet): MR1207673
    Digital Object Identifier: doi:10.1007/BF02097061
    Project Euclid: euclid.cmp/1104252313
  • Wilson, D. B. (2004). Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274–325.
    Mathematical Reviews (MathSciNet): MR2023023
    Zentralblatt MATH: 1040.60063
    Digital Object Identifier: doi:10.1214/aoap/1075828054
    Project Euclid: euclid.aoap/1075828054

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