On Markov Chains for Independent Sets

@article{Dyer2000OnMC,
  title={On Markov Chains for Independent Sets},
  author={Martin E. Dyer and Catherine S. Greenhill},
  journal={J. Algorithms},
  year={2000},
  volume={35},
  pages={17-49}
}
Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. In 1997, Luby and Vigoda described a rapidly mixing Markov chain for independent sets, which we refer to as the Luby?Vigoda chain. A new rapidly mixing Markov chain for independent sets is defined in this paper. Using path coupling, we obtain a polynomial upper bound for the mixing time of the new chain for a certain range of values of the parameter ?. This range is wider than the range… 

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References

SHOWING 1-10 OF 40 REFERENCES

Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

  • A. Sinclair
  • Computer Science
    Combinatorics, Probability and Computing
  • 1992
A new upper bound on the mixing rate is presented, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph, and improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system.

Fast convergence of the Glauber dynamics for sampling independent sets

This paper proves complementary hardness of approximation results, which show that it is hard to sample from this distribution when > c for a constant c > 0 and shows fast convergence of this dynamics.

COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS

By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state

A more rapidly mixing Markov chain for graph colorings

A new Markov chain is defined on k-colourings of graphs, and its convergence properties are related to the maximum degree ∆ of the graph, and it is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/Salas–Sokal chain in most circumstances.

Path coupling: A technique for proving rapid mixing in Markov chains

  • Russ BubleyM. Dyer
  • Mathematics
    Proceedings 38th Annual Symposium on Foundations of Computer Science
  • 1997
A new approach to the coupling technique, which is called path coupling, for bounding mixing rates, is illustrated, which may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method.

Exact sampling with coupled Markov chains and applications to statistical mechanics

This work describes a simple variant of this method that determines on its own when to stop and that outputs samples in exact accordance with the desired distribution, and uses couplings which have also played a role in other sampling schemes.

On Approximately Counting Colorings of Small Degree Graphs

A computer-assisted proof technique is used to establish rapid mixing of a new "heat bath" Markov chain on colorings using the method of path coupling and gives a general proof that the problem of exactly counting the number of proper k-colorings of graphs with maximum degree $\Delta$ is complete.

An interruptible algorithm for perfect sampling via Markov chains

A new algorithm is presented which again uses the same Markov chains to produce perfect samples from n, but is baaed on a different idea (namely, acceptance/rejection sampling); and eliminates user-impatience bias.

Markov Chain Algorithms for Planar Lattice Structures (Extended Abstract).

This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution.

The Markov chain Monte Carlo method: an approach to approximate counting and integration

The introduction of analytical tools with the aim of permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization.