• Corpus ID: 15734955

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

@inproceedings{Bubley1998BeatingT2,
title={Beating the 2$\Delta$ bound for approximately counting colourings: a computer-assisted proof of rapid mixing},
author={Russ Bubley and Martin E. Dyer and Catherine S. Greenhill},
booktitle={SODA '98},
year={1998}
}
• Published in SODA '98 1998
• Mathematics
We consider random walks on graph colourings of an nvertex graph. It has been shown by Jerrum and by Salas and Sokal that a simple random walk would mix rapidly provided the number of colours, k, exceeded the maximum degree A of the graph by a factor of at least 2. Lack of improvements on this bound led to a conjecture that k 2 2A was a natural barrier. We disprove this conjecture in the simplest case of 5-colouring graphs of maximum degree 3. Our proof involves a novel computer-assisted proof…

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## References

SHOWING 1-10 OF 32 REFERENCES
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.
Path coupling: A technique for proving rapid mixing in Markov chains
• 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.
Markov chains, computer proofs, and average-case analysis of best fit bin packing
• Computer Science
STOC '93
• 1993
A computer-aided approach to the analysis of higher-dimensional domains, using several open problems about the average-case behavior of the Best Fit bin packing algorithm as case studies and answering yes to the long-standing open question of whether there exist distributions of thii form for which Best Fit yields linearly-growing waste.
On Markov Chains for Independent Sets
• Mathematics
J. 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.
Approximating the Permanent
• Mathematics
SIAM J. Comput.
• 1989
A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...
A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph
• M. Jerrum
• Mathematics
Random Struct. Algorithms
• 1995
A fully polynomial randomized approximation scheme is presented for estimating the number of (vertex) k‐colorings of a graph of maximum degree Δ, when k ≥ 2Δ + 1. © 1995 John Wiley & Sons, Inc.
A random polynomial-time algorithm for approximating the volume of convex bodies
• Mathematics
JACM
• 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.
Random Generation of Combinatorial Structures from a Uniform Distribution
• Computer Science, Mathematics
Theor. Comput. Sci.
• 1986
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)
Approximation Algorithms for NP-Hard Problems
This book introduces unifying techniques in the analysis of approximation algorithms, intended for computer scientists and operations researchers interested in specific algorithm implementations, as well as design tools for algorithms.