# Path coupling using stopping times and counting independent sets and colorings in hypergraphs

@article{Bordewich2005PathCU, title={Path coupling using stopping times and counting independent sets and colorings in hypergraphs}, author={Magnus Bordewich and Martin E. Dyer and Marek Karpinski}, journal={Random Structures \& Algorithms}, year={2005}, volume={32} }

We analyse the mixing time of Markov chains using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Δ of a vertex and the minimum size m of an edge satisfy m ≥ 2Δ + 1. We also show that the Glauber dynamics for proper q‐colorings of a hypergraph mixes rapidly if m ≥ 4 and q > Δ, and if m = 3 and q ≥ 1.65Δ. We give related results on the hardness…

## 26 Citations

### Path Coupling Using Stopping Times

- MathematicsFCT
- 2005

It is shown that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Δ of a vertex and the minimum size m of an edge satisfy m ≥ 2Δ +1.65Δ.

### Metric Construction, Stopping Times and Path Coupling

- MathematicsElectron. Colloquium Comput. Complex.
- 2005

A stronger theorem for path coupling with stopping times is proved, using a metric which allows us to restrict analysis to standard one-step path coupling, providing insight for the design of nonstandard metrics giving improvements in the analysis of specific problems.

### Counting hypergraph colourings in the local lemma regime

- MathematicsSTOC
- 2018

This work gives a fully polynomial-time approximation scheme (FPTAS) to count the number of q-colorings for k-uniform hypergraphs with maximum degree Δ if k≥ 28 and q > 315Δ14/k−14 and gets the first approximate counting and sampling algorithms in the regime q≪Δ.

### Rapid mixing of hypergraph independent sets

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2019

We prove that the mixing time of the Glauber dynamics for sampling independent sets on n‐vertex k‐uniform hypergraphs is O(nlogn) when the maximum degree Δ satisfies Δ ≤ c2k/2, improving on the…

### On Sampling Simple Paths in Planar Graphs According to Their Lengths

- MathematicsMFCS
- 2015

This work considers the problem of sampling simple paths between two given vertices in a planar graph and proposes a natural Markov chain exploring such paths by means of “local” modifications, and shows that this chain is always ergodic and thus it converges to the desired sampling distribution for any planar graphs.

### Approximate Counting of Matchings in Sparse Uniform Hypergraphs

- MathematicsANALCO
- 2013

It is proved that it is NP-hard to approximate the number of matchings even for the class of 2-regular, linear, k-uniform hypergraphs, for all k ≥ 6, without the above restriction.

### Subset Glauber Dynamics on Graphs, Hypergraphs and Matroids of Bounded Tree-Width

- MathematicsElectron. J. Comb.
- 2014

It is shown that Glauber dynamics for a very wide class of polynomials mixes rapidly on graphs of bounded tree-width, including many cases in which the Glaubert dynamics does not mix rapidly for all graphs.

### Sampling Eulerian orientations of triangular lattice graphs

- MathematicsJ. Discrete Algorithms
- 2009

### A survey on the use of Markov chains to randomly sample colorings

- Mathematics
- 2006

Results on generating random colorings, and related technical improvements on the analysis of Markov chains for generating a random coloring of an input graph are surveyed.

### The complexity of approximately counting in 2-spin systems on k-uniform bounded-degree hypergraphs

- Mathematics, Computer ScienceInf. Comput.
- 2016

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