The mixing time of Glauber dynamics for coloring regular trees

Goldberg, Leslie Ann; Jerrum, Mark; Karpiński, Marek

doi:10.1002/rsa.20303

Cited by 10 publications

(31 citation statements)

References 24 publications

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“…Our result extends to more general k and b, thereby refining the general picture provided by [12] and [21]. …”

Section: Introductionsupporting

confidence: 80%

“…Our main result provides (nearly) sharp bounds on the mixing time and relaxation time of the Glauber dynamics for the complete tree, establishing a phase transition at the critical point k = b(1 + o b (1))/ ln b. Our proofs build upon the approaches used by [12] and [21]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 81%

“…Thus we want to establish a more precise picture than provided by the results of [12] and [21]. Our main result provides (nearly) sharp bounds on the mixing time and relaxation time of the Glauber dynamics for the complete tree, establishing a phase transition at the critical point k = b(1 + o b (1))/ ln b.…”

Section: Introductionmentioning

confidence: 88%

“…Subsequently, improved results were established for the tree. In particular, Goldberg et al [12] proved the mixing time is n Ω(b/(k ln b)) for the complete tree with branching factor b, and Lucier et al [21,22] proved the mixing time is n O(1+b/(k ln b)) for any tree with maximum degree b.…”

Section: Introductionmentioning

confidence: 99%

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Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Tetali¹,

Vera²,

Vigoda³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at k = b(1+o b (1))/ ln b. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for k = Cb/ ln b colors with constant C. For C ≥ 1 we prove the mixing time is O(n 1+o b (1) ln 2 n). On the other side, for C < 1 the mixing time experiences a slowing down, in particular, we prove it is O(n 1/C+o b (1) ln 2 n) andThe critical point C = 1 is interesting since it coincides (at least up to first order) to the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.

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“…Our result extends to more general k and b, thereby refining the general picture provided by [12] and [21]. …”

Section: Introductionsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Tetali¹,

Vera²,

Vigoda³

et al. 2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Independently, Goldberg, Jerrum and Karpinski [6] and Lucier and Molloy [13] showed a lower bound of n Ω(1+∆/k log ∆) on the mixing time for the case of the complete tree. Goldberg, Jerrum and Karpinski also showed an upper bound of n O(1+∆/ log ∆) for complete trees and constant ∆.…”

Section: Introductionmentioning

confidence: 99%

The Glauber Dynamics for Colourings of Bounded Degree Trees

Lucier¹,

Molloy²,

Peres

2009

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We study the Glauber dynamics Markov chain for k-colourings of trees with maximum degree ∆. For k ≥ 3, we show that the mixing time on every tree is at most n O(1+∆/(k log ∆)) . This bound is tight up to the constant factor in the exponent, as evidenced by the complete tree. Our proof uses a weighted canonical paths analysis and a variation of the block dynamics in which we exploit the differing relaxation times of blocks. *

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Randomly coloring planar graphs with fewer colors than the maximum degree

Hayes

Vera

Vigoda

2014

Random Struct. Alg.

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We study Markov chains for randomly sampling k‐colorings of a graph with maximum degree Δ. Our main result is a polynomial upper bound on the mixing time of the single‐site update chain known as the Glauber dynamics for planar graphs when k=Ω(Δ/logΔ). Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most Δ1−ϵ, for fixed ϵ>0. The main challenge when k≤Δ+1 is the possibility of “frozen” vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when Δ=O(1), even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using “local uniformity” properties. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 731–759, 2015

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The mixing time of Glauber dynamics for coloring regular trees

Cited by 10 publications

References 24 publications

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

Phase Transition for the Mixing Time of the Glauber Dynamics for Coloring Regular Trees

The Glauber Dynamics for Colourings of Bounded Degree Trees

Randomly coloring planar graphs with fewer colors than the maximum degree

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