Journal ArticleDOI
Analyzing Glauber dynamics by comparison of Markov chains
Dana Randall,Prasad Tetali +1 more
TLDR
This work uses the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single-point update algorithms are efficient, and relates the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed.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 nonlocal algorithms which have already been analyzed. This allows us to give polynomial time bounds for single-point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons. We also survey several other comparison techniques, along with specific applications, which have been used in the context of estimating mixing rates of Markov chains.read more
Citations
More filters
Book
Stochastic Network Optimization with Application to Communication and Queueing Systems
TL;DR: In this article, the authors present a modern theory of analysis, control, and optimization for dynamic networks, including wireless networks with time-varying channels, mobility, and randomly arriving traffic.
Book
Finite Markov Chains and Algorithmic Applications
TL;DR: 1. Basics of probability theory 2. Markov chains 3. Irreducible and aperiodic Markov Chains 4. Stationary distributions 5.Stationary distributions 6. Reversible Markov Chain Monte Carlo 7. Fast convergence of MCMC algorithms
Book
Mathematical Aspects of Mixing Times in Markov Chains
Ravi Montenegro,Prasad Tetali +1 more
TL;DR: The strength of the main techniques are illustrated by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.
Journal ArticleDOI
The Complexity of Counting in Sparse, Regular, and Planar Graphs
TL;DR: It is proved that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree.
Journal ArticleDOI
Mixing times of lozenge tiling and card shuffling Markov chains
TL;DR: In this article, the mixing times of a variety of Markov chains were bound by combining Fourier analysis with coupling arguments, and the mixing time was shown to be within a constant factor of their upper bounds.
References
More filters
Journal ArticleDOI
Geometric Bounds for Eigenvalues of Markov Chains
Persi Diaconis,Daniel Stroock +1 more
TL;DR: In this article, the second largest eigenvalue and spectral gap of a reversible Markov chain were derived for the random walk associated to approximate computation of the permanent. But these bounds depend on geometric quantities such as the maximum degree, diameter and covering number of associated graphs.
Journal ArticleDOI
Approximating the permanent
Mark Jerrum,Alistair Sinclair +1 more
TL;DR: A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...
Journal ArticleDOI
Approximate counting, uniform generation and rapidly mixing Markov chains
Alistair Sinclair,Mark Jerrum +1 more
TL;DR: In this article, it was shown that for self-reducible structures, almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible.
Journal ArticleDOI
Generating a random permutation with random transpositions
Journal ArticleDOI
Polynomial-time approximation algorithms for the Ising model
Mark Jerrum,Alistair Sinclair +1 more
TL;DR: A randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic Ising system to any specified degree of accuracy is presented.
Related Papers (5)
Approximate counting, uniform generation and rapidly mixing Markov chains
Alistair Sinclair,Mark Jerrum +1 more