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Randomized Algorithms
TLDR
These notes describe other important illustrations of randomized algo rithms in other areas of the theory of algorithms and describe some basic principles which typically underly the construction of randomized algorithms.
Path coupling: A technique for proving rapid mixing in Markov chains
  • Russ Bubley, M. Dyer
  • Mathematics
    Proceedings 38th Annual Symposium on Foundations…
  • 19 October 1997
TLDR
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.
Faster random generation of linear extensions
Graph orientations with no sink and an approximation for a hard case of #SAT
TLDR
All of the major combinatorial problems associated with sink-free graph orientations are considered: decision, construction, listing, counting, approximate counting and approximate sampling.
Path Coupling, Dobrushin Uniqueness, and Approximate Counting
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
TLDR
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.
Randomized algorithms - approximation, generation and counting
  • Russ Bubley
  • Mathematics
    Distinguished dissertations
  • 1 March 2000
TLDR
Randomized Al algorithms discusses two problems of fine pedigree: counting and generation, both of which are of fundamental importance to discrete mathematics and probability, and uses the technique of coupling before introducing "path coupling" a new technique which radically simplifies and improves upon previous methods in the area.
An elementary analysis of a procedure for sampling points in a convex body
TLDR
A new method for proving the convergence of an algorithm for sampling almost uniformly at random from a convex body in high dimension by using a more elementary coupling argument.
On Approximately Counting Colorings of Small Degree Graphs
TLDR
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.
A New Approach to Polynomial-Time Generation of Random Points in Convex Bodies
In this paper we describe a new method for proving the polynomial-time convergence of an algorithm for sampling (almost) uniformly at random from a convex body in high dimension. Previous approaches
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