Analyzing Glauber dynamics by comparison of Markov chains

Randall, Dana; Tetali, Prasad

doi:10.1007/bfb0054330

Cited by 50 publications

(77 citation statements)

References 17 publications

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“…In the adapted setting, if C i is a single vertex then this corresponds to having all adjacent edges colored the same. We use path coupling to prove that a related chain M E mixes rapidly and then use the comparison technique (see, [6,20]) to relate the mixing time of M E to the mixing time of M C . The chain M E is a generalization of the edge chain introduced by Luby and Vigoda [18].…”

Section: Finding a Conforming 2-coloringmentioning

confidence: 99%

“…To prove that M C is rapidly mixing, we first use path coupling to prove that a related chain M E mixes rapidly, and then use the comparison technique (see, [6,20] Starting at any initial conforming coloring, iterate the following:…”

Section: Rapid Mixing Of M Cmentioning

confidence: 99%

“…Let P ′ and P be two reversible Markov chains with mixing times τ (ϵ) and τ ′ (ϵ) respectively on the same state space Ω with the same stationary distribution π . The comparison method (see [6] and [20]) allows us to relate the mixing times of these two chains. Let E(P ) = {(x, y) : P (x, y) > 0} and E(P ′ ) = {(x, y) : P ′ (x, y) > 0} denote the sets of edges of the two graphs, viewed as directed graphs.…”

Section: Rapid Mixing Of M Cmentioning

confidence: 99%

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Algorithms to approximately count and sample conforming colorings of graphs

Miracle

Randall

2016

Discrete Applied Mathematics

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a b s t r a c tGiven a multigraph G and a function F that assigns a forbidden ordered pair of colors to each edge e, we say a coloring C of the vertices is conforming to F if for all e = (u, v), (C(u), C (v)) ̸ = F (e). Conforming colorings generalize many natural graph theoretic concepts, including independent sets, vertex colorings, list colorings, H-colorings and adapted colorings and consequently there are known complexity barriers to sampling and counting. We introduce natural Markov chains on the set of conforming colorings and provide general conditions for when they can be used to design efficient Monte Carlo algorithms for sampling and approximate counting.

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Section: Finding a Conforming 2-coloringmentioning

confidence: 99%

Section: Rapid Mixing Of M Cmentioning

confidence: 99%

Section: Rapid Mixing Of M Cmentioning

confidence: 99%

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Algorithms to approximately count and sample conforming colorings of graphs

Miracle

Randall

2016

Discrete Applied Mathematics

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“…We will use the formulation of the Diaconis and Saloff-Coste result from [22], restated here for convenience. Note that we are using E(P ) to denote the set of edges corresponding to moves between adjacent states in the Markov chain with transition matrix P .…”

Section: Comparison Of τ M and τ M Tmentioning

confidence: 99%

“…(See[22, Proposition 4].) Suppose P and P are the transition matrices of two reversible Markov chains, M and M, both…”

mentioning

confidence: 99%

Sampling Eulerian orientations of triangular lattice graphs

Creed

2009

Journal of Discrete Algorithms

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We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although Mihail and Winkler (1989) showed that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar graph which moves between orientations by reversing the edges of directed faces. Using path coupling and the comparison method we obtain a polynomial upper bound on the mixing time of this chain for any solid subgraph of the triangular lattice. By considering the conductance of the chain we show that there exist non-solid subgraphs (subgraphs with holes) for which the chain will always take an exponential amount of time to converge. Finally, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs (Mihail and Winkler had already established this for general graphs).

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Random dyadic tilings of the unit square

Janson

Randall

Spencer

2002

Random Struct Algorithms

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ABSTRACT:A "dyadic rectangle" is a set of the form R ϭ [a2Ϫt ], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2 n nonoverlapping dyadic rectangles, each of area 2Ϫn , whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.

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Analyzing Glauber dynamics by comparison of Markov chains

Cited by 50 publications

References 17 publications

Algorithms to approximately count and sample conforming colorings of graphs

Algorithms to approximately count and sample conforming colorings of graphs

Sampling Eulerian orientations of triangular lattice graphs

Random dyadic tilings of the unit square

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