Abstract
A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or “Glauber dynamics.” Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems.
In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating tilings, colorings and independent sets.
Research supported by NSF Grants No. CCR-9703206 and CCR-9503952.
Preview
Unable to display preview. Download preview PDF.
References
Aldous, D. Random walks on finite groups and rapidly mixing Markov chains. Séminaire de Probabilités XVII, 1981/82, Springer Lecture Notes in Mathematics 986, pp. 243–297.
Baxter, R.J. Exactly solved models in statistical mechanics. Academic Press, London, 1982.
van den Berg, J. and Steif, J.E. Percolation and the hard-core lattice gas model. Stochastic Processes and their Applications 49, 1994, pp. 179–197.
Diaconis, P. and Stroock, D. Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probability 1, 1991, pp. 36–61.
Diaconis, P. and Saloff-Coste, L. Comparison theorems for reversible Markov chains. Ann. Appl. Probability 3, 1993, pp. 696–730.
Diaconis, P. and Saloff-Coste, L. Logarithmic Sobolev Inequalities for Finite Markov Chains. Ann. of Appl. Probab. 6 (1996), pp. 695–750.
Jerrum, M. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures and Algorithms 7, 1995, pp. 157–165.
Jerrum, M.R. and Sinclair, A.J. Approximating the permanent. SIAM Journal on Computing 18 (1989), pp. 1149–1178.
Jerrum, M., Valiant, L. and Vazirani, V. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science 43 (1986), pp. 169–188.
Kannan, R., Tetali, and P., Vempala, S. Simple Markov chain algorithms for generating bipartite graphs and tournaments. Proc. of the 8th ACM-SIAM Symp. on Discrete Algorithms January 1997.
Lieb, E.H. Residual entropy of square ice. Physical Review 162, 1967, pp. 162–172.
Lubin, M. and Sokal A.D. Comment on “Antiferromagnetic Potts Model. Phys. Rev. Lett. 71, 1993, pp. 17–78.
Luby, M., Randall, D. and Sinclair, A. Markov Chain Algorithms for Planar Lattice Structures. Proc. 36th IEEE Symposium on Foundations of Computing (1995), pp. 150–159.
Luby, M., Vigoda, E. Approximately counting up to four. Proc. 29th ACM Symposium on Theory of Computing (1997), pp. 150–159.
Madras, N. and Randall, D. Factoring Graphs to Bound Mixing Time. Proc. 37th IEEE Symposium on Foundations of Computing (1996).
Sinclair, A.J. and Jerrum, M.R. Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Computation 82 (1989), pp. 93–133.
Sinclair, A.J. Algorithms for random generation & counting: a Markov chain approach. Birkhäuser, Boston, 1993, pp. 47–48.
Sinclair, A.J. Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability, & Computing. 1 (1992), pp. 351–370.
Thurston, W. Conway's tiling groups. American Mathematical Monthly 97, 1990, pp. 757–773.
Vigoda, E. personal communication.
Welsh, D.J.A. The computational complexity of some classical problems from statistical physics. In Disorder in Physical Systems, (G. Grimmett and D. Welsh eds.). Claredon Press, Oxford, 1990, pp. 323–335.
Welsh, D.J.A. Approximate counting. In Surveys in Combinatorics, (R.A. Bailey, ed.). Cambridge University Press, London Math Society Lecture Notes 241, 1997, pp. 287–317.
Wilson, D.B. Mixing times of lozenge tiling and card shuffling Markov chains, draft of manuscript.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Randall, D., Tetali, P. (1998). Analyzing Glauber dynamics by comparison of Markov chains. In: Lucchesi, C.L., Moura, A.V. (eds) LATIN'98: Theoretical Informatics. LATIN 1998. Lecture Notes in Computer Science, vol 1380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054330
Download citation
DOI: https://doi.org/10.1007/BFb0054330
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64275-6
Online ISBN: 978-3-540-69715-2
eBook Packages: Springer Book Archive