# 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}
}
• Published 6 January 2005
• Mathematics
• Random Structures & Algorithms
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…

### Path Coupling Using Stopping Times

• Mathematics
FCT
• 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

• Mathematics
Electron. 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

• Mathematics
STOC
• 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 Science
Random 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

• Mathematics
MFCS
• 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

• Mathematics
ANALCO
• 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

• Mathematics
Electron. 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.

### 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.

## References

SHOWING 1-10 OF 37 REFERENCES

### Very rapidly mixing Markov Chains for 2D-colorings and for independent sets in a graph with maximum degree 4

We introduce a new technique for analyzing the mixing rate of Markov chains. We use it to prove that the Glauber dynamics on 2Δ-colorings of a graph with maximum degree Δ mixes in O(n log n) time. We

### A more rapidly mixing Markov chain for graph colorings

• Mathematics
Random Struct. Algorithms
• 1998
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.

### On Approximately Counting Colorings of Small Degree Graphs

• Mathematics
SIAM J. Comput.
• 1999
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 Extension of Path Coupling and Its Application to the Glauber Dynamics for Graph Colorings

• Mathematics
SIAM 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 $\eta$.

### The complexity of counting colourings and independent sets in sparse graphs and hypergraphs

Using polynomial interpolation techniques, it is shown that certain counting problems involving colourings of graphs and independent sets in hypergraphs are #P-complete and efficient approximate counting is the most one can realistically expect to achieve.

### Approximating coloring and maximum independent sets in 3-uniform hypergraphs

• Mathematics
SODA '01
• 2001
Two approximation algorithms for the coloring problem and the maximum independent set problem in 3-uniform hypergraphs are discussed and results are obtained through semidefinite programming relaxations of these optimization problems.

### Approximating Maximum Independent Sets in Uniform Hypergraphs

• Mathematics, Computer Science
MFCS
• 1998
For fixed k≥2, the problems of approximating the independence number and the chromatic number of k-uniform hypergraphs on n vertices are considered and for both problems polynomial time approximation algorithms with approximation ratios O(n/(log(k-1) n)2) are described.

### Fast convergence of the Glauber dynamics for sampling independent sets

• Mathematics
Random Struct. Algorithms
• 1999
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.

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

• Mathematics
SODA '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.

### Randomly coloring constant degree graphs

• Physics, Mathematics
45th Annual IEEE Symposium on Foundations of Computer Science
• 2004
It is proved that the Glauber dynamics converges to a random coloring after O(n log n) steps assuming k /spl ges/ k/ sub 0/ for some absolute constant k/sub 0/,.