We prove that any Markov chain that performs local, reversible updates on randomly chosen vertices of a bounded-degree graph necessarily has mixing time at least Ω(n logn), where n is the number of vertices. Our bound applies to the so-called “Glauber dynamics” that has been used extensively in algorithms for the Ising model, independent sets, graph colorings and other structures in computer science and statistical physics, and demonstrates that many of these algorithms are optimal up to constant factors within their class. Previously, no superlinear lower bound was known for this class of algorithms. Though widely conjectured, such a bound had been proved previously only in very restricted circumstances, such as for the empty graph and the path. We also show that the assumption of bounded degree is necessary by giving a family of dynamics on graphs of unbounded degree with mixing time O(n).
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
References
Achlioptas, D., Molloy, M., Moore, C. and van Bussel, F. (2004). Sampling grid colorings with fewer colors. LATIN 2004: Theoretical Informatics. Lecture Notes in Comput. Sci. 2976 80--89. Springer, Berlin.
Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. Séminaire de Probabilites XVII. Lecture Notes in Math. 986 243--297. Springer, Berlin.
Aldous, D. and Fill, J. (2002). Reversible Markov chains and random walks on graphs. Unpublished manuscript. Available at www.stat.berkeley.edu/users/aldous/RWG/book.html. (References are to the September 10, 2002 version of Chapter 3.)
Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. Ser. B 36 192--236.
Borgs, C., Chayes, J. T., Frieze, A. M., Kim, J. H., Tetali, P., Vigoda, E. and Vu, V. H. (1999). Torpid mixing of some Monte Carlo Markov chain algorithms from statistical physics. In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science 218--229. IEEE Computer Soc., Los Alamitos, CA.
Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159--179.
Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695--750.
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 colourings. SIAM J. Comput. 30 1962--1975.
Dyer, M., Goldberg, L. A. and Jerrum, M. (2006). Systematic scan for sampling colourings. Ann. Appl. Probab. 16 185--230.
Dyer, M., Frieze, A. M. and Jerrum, M. (2002). On counting independent sets in sparse graphs. SIAM J. Comput. 31 1527--1541.
Dyer, M., Sinclair, A., Vigoda, E. and Weitz, D. (2004). Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures Algorithms 24 461--479.
Galvin, D. and Tetali, P. (2006). Slow mixing of the Glauber dynamics for the hard-core model on regular bipartite graphs. Random Structures Algorithms 28 427--443.
Jerrum, M. R. (1995). A very simple algorithm for estimating the number of $k$-colourings of a low-degree graph. Random Structures Algorithms 7 157--165.
Martinelli, F. and Olivieri, E. (1994). Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Comm. Math. Phys. 161 447--486.
Moussouris, J. (1974). Gibbs and Markov random systems with constraints. J. Statist. Phys. 10 11--33.
Vigoda, E. (2000). Improved bounds for sampling colorings. J. Math. Phys. 41 1555--1569.
Vigoda, E. (2001). A note on the Glauber dynamics for sampling independent sets. Electron. J. Combin. 8.
