Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

Chen, Zongchen; Liu, Kuikui; Vigoda, Eric

doi:10.1109/focs46700.2020.00124

Cited by 37 publications

(54 citation statements)

References 20 publications

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“…Therefore, the Gibbs distribution

π

is the stationary distribution of the chain. Indeed, if Z′ is obtained from Z by deleting a vertex v then

P (Z, Z^{'}) = \frac{1}{n (1 + λ)} and P (Z^{'}, Z) = \frac{λ}{n (1 + λ)} .

Very recently, Chen et al [14] proved that the Glauber dynamics converges in time

O (n^{2 + 32 / δ})

whenever

λ \leq (1 - δ) λ_{c}

, where

Δ

is the maximum vertex degree and

λ_{c} = {(Δ - 1)}^{Δ - 1} / {(Δ - 2)}^{Δ}

is the tree uniqueness threshold. Now, by an easy calculation,

λ < e / n

implies that

λ < λ_{c}

.…”

Section: Markov Chainsmentioning

confidence: 99%

“…We would like to thank the anonymous referee for their helpful comments and for bringing [3, 14] to our attention. We also thank Ferenc Bencs for letting us know that the results of [8] lead to an FPTAS for the independence polynomial of bounded‐degree fork‐free graphs.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Very recently (after this paper was submitted), Anari et al [3] proved that the Glauber dynamics for weighted independent sets (the hardcore model) is rapidly mixing in the tree uniqueness region, and their analysis was simplified and generalized by Chen et al [14], improving the mixing time.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Counting independent sets in graphs with bounded bipartite pathwidth

Dyer

Greenhill

Müller

2021

Random Struct Algorithms

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We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalization of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw‐free graphs, which generalize line graphs. We consider two further generalizations of claw‐free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially bounded vertex weights.

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“…Therefore, the Gibbs distribution

π

is the stationary distribution of the chain. Indeed, if Z′ is obtained from Z by deleting a vertex v then

P (Z, Z^{'}) = \frac{1}{n (1 + λ)} and P (Z^{'}, Z) = \frac{λ}{n (1 + λ)} .

Very recently, Chen et al [14] proved that the Glauber dynamics converges in time

O (n^{2 + 32 / δ})

whenever

λ \leq (1 - δ) λ_{c}

, where

Δ

is the maximum vertex degree and

λ_{c} = {(Δ - 1)}^{Δ - 1} / {(Δ - 2)}^{Δ}

is the tree uniqueness threshold. Now, by an easy calculation,

λ < e / n

implies that

λ < λ_{c}

.…”

Section: Markov Chainsmentioning

confidence: 99%

Section: Acknowledgmentsmentioning

confidence: 99%

See 1 more Smart Citation

Counting independent sets in graphs with bounded bipartite pathwidth

Dyer

Greenhill

Müller

2021

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

“…The phrase "rapidly mixing" indicates that the dynamics converges close to the Gibbs distribution in a number of updates that is polynomial in the size of the graph. In a recent paper, Chen, Liu and Vigoda [6] show that Glauber dynamics mixes rapidly up to a natural threshold for β, depending on the maximum degree of the graph, beyond which it was already known that the mixing time is exponential. Here we show that, by suitably restricting the underlying graph, we can have rapid mixing for all β.…”

Section: Introductionmentioning

confidence: 99%

Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

Dyer

Heinrich

Jerrum

et al. 2021

Combinator. Probab. Comp.

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We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

show abstract

“…Despite this clear complexity picture, prior to the introduction of spectral independence, the algorithms for λ < λ c (d) were based on elaborate enumeration techniques whose running times scale as n O(log d) [Wei06, LLY13, PR17, PR19]. The analysis of Glauber dynamics 1 using spectral independence in the regime λ < λ c (d) yielded initially n O(1) algorithms for any d [ALG20], and then O(n log n) for bounded-degree graphs [CLV21] (see also [CLV20]). More recently, Chen, Feng, Yin, and Zhang [CFYX21] obtained O(n 2 log n) results for arbitrary graphs G = (V, E) that apply when λ < λ c (∆ G − 1), where ∆ G is the maximum degree of G (see also [JPV21] for related results when ∆ G grows like log n).…”

Section: Introductionmentioning

confidence: 99%

Fast sampling via spectral independence beyond bounded-degree graphs

Bezáková¹,

Galanis²,

Goldberg³

et al. 2021

Preprint

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Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations.Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs.As a main application of our techniques, we consider the random graph G(n, d/n), where the previously known algorithms run in time n O(log d) or applied only to large d. We refine these algorithmic bounds significantly, and develop fast n 1+o(1) algorithms based on Glauber dynamics that apply to all d, throughout the uniqueness regime.

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Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

Cited by 37 publications

References 20 publications

Counting independent sets in graphs with bounded bipartite pathwidth

Counting independent sets in graphs with bounded bipartite pathwidth

Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

Fast sampling via spectral independence beyond bounded-degree graphs

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