## 187 Citations

### The Mixing of Markov Chains on Linear Extensions in Practice

- MathematicsIJCAI
- 2017

The empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive bounds or termination rules.

### A Sequential Importance Sampling Algorithm for Counting Linear Extensions

- Mathematics, Computer ScienceACM J. Exp. Algorithmics
- 2020

This technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method, and defines and reports performance on two importance functions.

### Surveys in Combinatorics, 1999: Random Walks on Combinatorial Objects

- Mathematics
- 1999

Approximate sampling from combinatorially-defined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated…

### Upper bounds on mixing time of finite Markov chains

- Mathematics
- 2020

. We provide a general framework for computing upper bounds on mixing times of ﬁnite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and…

### Spectral Gap for Random-to-Random Shuffling on Linear Extensions

- Mathematics, Computer ScienceExp. Math.
- 2017

This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by n2/(n + 2) and a mixing time of O(n2log n) as for the usual random-to-random shuffling.

### Mixing times of lozenge tiling and card shuffling Markov chains

- Mathematics
- 2004

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its…

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

- MathematicsProceedings 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.

### An approximation algorithm for random generation of capacities

- Computer Science, MathematicsArXiv
- 2022

This paper proposes the 2-layer approximation method, which generates a subset of linear extensions, eliminating those with very low probability, and shows that this method has similar performance compared to the Markov chain but is much less time consuming.

### L ∞ -Discrepancy Analysis of Polynomial-Time Deterministic Samplers Emulating Rapidly Mixing Chains

- Mathematics, Computer ScienceCOCOON
- 2014

A deterministic sampling algorithm based on deterministic random walk, such as the rotor-router model (a.k.a. Propp machine) is proposed, which provides samples with a “distribution” with a point-wise distance at most e from the target distribution, in time polynomial in the input size and e − 1.

## References

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### Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains

- MathematicsWG
- 1987

The general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour.

### Markov chains for linear extensions, the two-dimensional case

- MathematicsSODA '97
- 1997

It is shown that monotone coupling from the past can be applied in the case of linear extensions of two-dimensional orders using Markou chains and for width two orders a mixing rate of O(n3 logn) is proved.

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

- MathematicsProceedings 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.

### Counting linear extensions

- Mathematics
- 1991

We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is…

### On Markov Chains for Independent Sets

- MathematicsJ. Algorithms
- 2000

A new rapidly mixing Markov chain for independent sets is defined and a polynomial upper bound for the mixing time of the new chain is obtained for a certain range of values of the parameter ?, which is wider than the range for which the mixingTime of the Luby?Vigoda chain is known to be polynomially bounded.

### Exact sampling with coupled Markov chains and applications to statistical mechanics

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 1996

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 the conductance of order Markov chains

- Mathematics
- 1991

Let Q be a convex solid in ℝn, partitioned into two volumes u and v by an area s. We show that s>min(u,v)/diam Q, and use this inequality to obtain the lower bound n-5/2 on the conductance of order…

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

- Computer Science
- 1996

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.

### Generating Linear Extensions Fast

- Computer Science, MathematicsSIAM J. Comput.
- 1994

The algorithm presented here is the first constant amortized time algorithm for generating a "naturally defined" class of combinatorial objects for which the corresponding counting problem is #P-complete.

### 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.