Source: Ann. Appl. Probab.
Volume 16, Number 3
We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach.
As an application, we show that for k/Δ>1.764, the Glauber dynamics on k-colorings of a graph on n vertices with maximum degree Δ converges in O(nlog n) steps, assuming Δ=Ω(log n) and that the graph is triangle-free. Previously, girth ≥5 was needed.
As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at fugacity λ<(1−ɛ)e/Δ, on a regular graph G on n vertices of degree Δ=Ω(log n) and girth ≥6. The best known algorithm for general graphs currently assumes λ<2/(Δ−2).
Aldous, D. J. (1983). Random walks on finite groups and rapidly mixing Markov chains. Séminaire de Probabilities XVII. Lecture Notes in Math. 986 243--297. Springer, Berlin.
van den Berg, J. and Steif, J. E. (1994). Percolation and the hard-core lattice gas model. Stochastic Process. Appl. 49 179--197.
Bubley, R. and Dyer, M. E. (1997). Path coupling: A technique for proving rapid mixing in Markov chains. In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science 223--231. IEEE Computer Society Press, Los Alamitos, CA.
Doeblin, W. (1938). Exposé de la théorie des chaînes simples constantes de Markov à un nombre fini d'états. Revue Mathématique de l'Union Interbalkanique 2 77--105.
Dyer, M. and Frieze, A. (2003). Randomly colouring graphs with lower bounds on girth and maximum degree. Random Structures Algorithms 23 167--179.
Dyer, M., Frieze, A., Hayes, T. and Vigoda, E. (2004). Randomly coloring constant degree graphs. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science 582--589. IEEE Computer Society Press, Los Alamitos, CA.
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.
Goldberg, L. A., Martin, R. and Paterson, M. (2004). Strong spatial mixing for graphs with fewer colours. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science 562--571. IEEE Computer Society Press, Los Alamitos, CA.
Hayes, T. P. (2003). Randomly coloring graphs of girth at least five. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing 269--278. ACM, New York.
Hayes, T. P. and Vigoda, E. (2003). A non-Markovian coupling for randomly sampling colorings. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science 618--627. IEEE Computer Society Press, Los Alamitos, CA.
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.
Jerrum, M. R., Sinclair, A. and Vigoda, E. (2004). A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. Assoc. Comput. Machinery 51 671--697.
Kannan, R., Lovász, L. and Simonovits, M. (1997). Random walks and an $O^*(n^5)$ volume algorithm for convex bodies. Random Structures Algorithms 11 1--50.
Lindvall, T. (2002). Lectures on the Coupling Method. Dover Publications Inc., New York.
Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1717 93--191. Springer, Berlin.
Molloy, M. (2004). The Glauber dynamics on colorings of a graph with high girth and maximum degree. SIAM J. Comput. 33 712--737.
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 1--8.