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

@article{Bubley1997PathCA, title={Path coupling: A technique for proving rapid mixing in Markov chains}, author={Russ Bubley and Martin E. Dyer}, journal={Proceedings 38th Annual Symposium on Foundations of Computer Science}, year={1997}, pages={223-231} }

The main technique used in algorithm design for approximating #P-hard counting problems is the Markov chain Monte Carlo method. At the heart of the method is the study of the convergence (mixing) rates of particular Markov chains of interest. In this paper we illustrate a new approach to the coupling technique, which we call path coupling, for bounding mixing rates. Previous applications of coupling have required detailed insights into the combinatorics of the problem at hand, and this…

## 373 Citations

### Rapidly Mixing Markov Chains: A Comparison of Techniques (A Survey)

- MathematicsArXiv
- 2016

It is proved that for a large class of Markov chains canonical paths does capture rapid mixing, which motivates the question of whether Coupling can be made to work whenever the chain is rapidly mixing.

### Canonical Paths for MCMC: from Art to Science

- Computer ScienceSODA
- 2016

The task of designing canonical paths for MCMC is reduced to solving a set of linear equations, which can be automatically done even by a machine, and fully polynomial-time randomized approximation schemes (FPRAS) for counting the number of b-matching with b ≤ 7 and b-edge-cover with b ≥ 2 are obtained.

### Rapid mixing through decomposition and induction

- Mathematics
- 2004

The thesis concludes with the treatment of a novel decomposition method and its application to bounding the convergence rate of a random process on finite, complete d-ary trees.

### An Extension of Path Coupling and Its Application to the Glauber Dynamics for Graph Colorings

- MathematicsSIAM J. Comput.
- 2000

It is shown that the Glauber dynamics has O(n log(n) mixing time for triangle-free $\Delta$-regular graphs if k colors are used, where $k\geq (2-\eta)\Delta$, for some small positive constant $\eta$.

### Disjoint Decomposition of Markov Chains and Sampling Circuits in Cayley Graphs

- MathematicsCombinatorics, Probability and Computing
- 2006

A new version of the decomposition theorem is presented where the pieces partition the state space, rather than forming a cover where pieces overlap, as was previously required, which is more natural and better suited to many applications.

### Delayed path coupling and generating random permutations

- Computer Science, MathematicsRandom Struct. Algorithms
- 2000

The main result is that distributed mixing needs log n simple point-to-point communication rounds to generate a permutation almost uniformly at random.

### Approximating graph-theoretic counting problems with Markov chain simulation

- Mathematics
- 2018

In graph theory, it is hard to find exact answers for certain counting problems efficiently. For example, counting the number of proper k-colourings or independent sets on a graph are hard to do…

### Perfect sampling using bounding chains

- Mathematics
- 1999

In Monte Carlo simulation, samples are drawn from a distribution to estimate properties of the distribution that are too difficult to compute analytically. This has applications in numerous fields,…

### Markov chains for sampling matchings

- Mathematics
- 2008

A lemma is developed which obtains better bounds on the mixing time in this case than existing theorems, in the case where β = 1 and the probability of a change in distance is proportional to the distance between the two states.

### Fast sampling and counting 𝑘-SAT solutions in the local lemma regime

- MathematicsSTOC
- 2020

The main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well developed MCMC method applicable on disconnected state spaces such as SAT solutions.

## References

SHOWING 1-10 OF 20 REFERENCES

### Path Coupling, Dobrushin Uniqueness, and Approximate Counting

- Mathematics
- 1997

In this paper we illustrate new techniques for bounding the mixing rates of certain Markov chains. Whereas previous techniques have required extensive insight into the combinatorics of the problem in…

### Beating the 2Δ bound for approximately counting colourings: a computer-assisted proof of rapid mixing

- MathematicsSODA '98
- 1998

This work proposes a novel computer-assisted proof technique to establish rapid mixing of a new “heat bath” Markov chain on colourings using the method of path coupling and outlines an extension to 7-colourings of triangle-free 4-regular graphs.

### Polynomial Time Randomised Approximation Schemes for Tutte-Gröthendieck Invariants: The Dense Case

- MathematicsElectron. Colloquium Comput. Complex.
- 1994

A general technique is developed that supplies fully polynomial randomised approximation schemes for approximating the value of T(G; x, y) for any dense graph G, that is, any graph on n vertices whose minimum.

### Graph orientations with no sink and an approximation for a hard case of #SAT

- MathematicsSODA '97
- 1997

All of the major combinatorial problems associated with sink-free graph orientations are considered: decision, construction, listing, counting, approximate counting and approximate sampling.

### Monte-Carlo algorithms for enumeration and reliability problems

- Mathematics24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
- 1983

A simple but very general Monte-Carlo technique for the approximate solution of enumeration and reliability problems and several applications are given.

### A random polynomial-time algorithm for approximating the volume of convex bodies

- MathematicsJACM
- 1989

The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within K within Euclidean space.

### Approximately counting up to four (extended abstract)

- Mathematics, Computer ScienceSTOC '97
- 1997

We presenta fully-polynomial scheme to approximate the number of independentsets in graphswith maximum degree four. In general, for graphswithmaximum degree A ~ 4, the scheme approximates a weighted…

### Random walks on finite groups and rapidly mixing markov chains

- Mathematics
- 1983

© Springer-Verlag, Berlin Heidelberg New York, 1983, tous droits reserves. L’acces aux archives du seminaire de probabilites (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml),…

### Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

- Mathematics
- 1997

We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature)…