Randomly coloring planar graphs with fewer colors than the maximum degree

Hayes, Thomas P.; Vera, Juan C.; Vigoda, Eric

doi:10.1002/rsa.20560

“…They showed O ( n log n ) mixing time for k ≥Δ +3, even for any boundary condition which is a fixed coloring of the leaves of the complete tree. For planar graphs with maximum degree Δ, Hayes et al [10] were able to achieve polynomial mixing time for k ≪Δ, in particular, they showed polynomial mixing time when k > 100Δ/log Δ. More recently, Tetali et al [24] have shown that on the complete tree the mixing time of the Glauber dynamics has a phase transition at k ≈︁Δ /log Δ.…”

Section: Introductionmentioning

confidence: 99%

Randomly coloring constant degree graphs

Dyer

¹

,

Frieze

²

,

Hayes

³

et al. 2012

Random Struct Algorithms

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We study a simple Markov chain, known as the Glauber dynamics, for generating a random k ‐coloring of an n ‐vertex graph with maximum degree Δ. We prove that, for every ε > 0, the dynamics converges to a random coloring within O(nlog n) steps assuming k ≥ k0(ε) and either: (i) k/Δ > α* + ε where α*≈︁ 1.763 and the girth g ≥ 5, or (ii) k/Δ >β * + ε where β*≈︁ 1.489 and the girth g ≥ 7. Our work improves upon, and builds on, previous results which have similar restrictions on k/Δ and the minimum girth but also required Δ = Ω (log n). The best known result for general graphs is O(nlog n) mixing time when k/Δ > 2 and O(n2) mixing time when k/Δ > 11/6. Related results of Goldberg et al apply when k/Δ > α* for all Δ ≥ 3 on triangle‐free “neighborhood‐amenable” graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

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“…There are previous works [11,12] which utilize the spectral radius of the adjacency matrix of the input graph G to design a suitable distance function for path coupling. In contrast, we use insights from the analysis of the BP operator to derive a suitable distance function.…”

Section: Contributionsmentioning

confidence: 99%

“…The fact that ω * is a Jacobian attractive fixpoint implies the existence of a nonnegative Φ withĴΦ < Φ. Thus, the theorem would follow immediately if the spectral radius ofĴ is ρ(Ĵ ) ≤ 1 − δ/6 and J has a principal eigenvector with each entry from the bounded range [1,12]. However, explicitly calculating this principal eigenvector can be challenging on general graphs.…”

Section: Path Coupling Distance Functionmentioning

confidence: 99%

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Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

Efthymiou¹,

Hayes²,

et al. 2019

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We study the hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter (aka fugacity) λ > 0. For constant ∆, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree ∆ when λ < λ c (∆). Sly (2010) showed that there is no FPRAS, unless NP=RP, when λ > λ c (∆). The threshold λ c (∆) is the critical point for the statistical physics phase transition for uniqueness/non-uniqueness on the infinite ∆-regular tree. The running time of Weitz's algorithm is exponential in log ∆. Here we present an FPRAS for the partition function whose running time is O * (n 2 ). We analyze the simple single-site Markov chain known as the Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant ∆ 0 such that for all graphs with maximum degree ∆ ≥ ∆ 0 and girth ≥ 7 (i.e., no cycles of length ≤ 6), the mixing time of the Glauber dynamics is O(n log n) when λ < λ c (∆). Our work complements that of Weitz which applies for small constant ∆ whereas our work applies for all ∆ at least a sufficiently large constant ∆ 0 (this includes ∆ depending on n = |V |).Our proof utilizes loopy BP (belief propagation) which is a widely-used algorithm for inference in graphical models. A novel aspect of our work is using the principal eigenvector for the BP operator to design a distance function which contracts in expectation for pairs of states that behave like the BP fixed point. We also prove that the Glauber dynamics behaves locally like loopy BP. As a byproduct we obtain that the Glauber dynamics, after a short burn-in period, converges close to the BP fixed point, and this implies that the fixed point of loopy BP is a close approximation to the Gibbs distribution. Using these connections we establish that loopy BP quickly converges to the Gibbs distribution when the girth ≥ 6 and λ < λ c (∆).

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“…Thus, the theorem would follow immediately if the spectral radius ofĴ is ρ(Ĵ ) ≤ 1 − δ/6 and J has a principal eigenvector with each entry from the bounded range [1,12]. However, explicitly calculating this principal eigenvector can be challenging on general graphs.…”

Section: Path Coupling Distance Functionmentioning

confidence: 99%

Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

Efthymiou¹,

Hayes

²

,

Štefankovič

³

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We study the hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter (aka fugacity) λ > 0. For constant ∆, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree ∆ when λ < λ c (∆). Sly (2010) showed that there is no FPRAS, unless NP=RP, when λ > λ c (∆). The threshold λ c (∆) is the critical point for the statistical physics phase transition for uniqueness/non-uniqueness on the infinite ∆-regular tree. The running time of Weitz's algorithm is exponential in log ∆. Here we present an FPRAS for the partition function whose running time is O * (n 2 ). We analyze the simple single-site Markov chain known as the Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant ∆ 0 such that for all graphs with maximum degree ∆ ≥ ∆ 0 and girth ≥ 7 (i.e., no cycles of length ≤ 6), the mixing time of the Glauber dynamics is O(n log n) when λ < λ c (∆). Our work complements that of Weitz which applies for small constant ∆ whereas our work applies for all ∆ at least a sufficiently large constant ∆ 0 (this includes ∆ depending on n = |V |).Our proof utilizes loopy BP (belief propagation) which is a widely-used algorithm for inference in graphical models. A novel aspect of our work is using the principal eigenvector for the BP operator to design a distance function which contracts in expectation for pairs of states that behave like the BP fixed point. We also prove that the Glauber dynamics behaves locally like loopy BP. As a byproduct we obtain that the Glauber dynamics, after a short burn-in period, converges close to the BP fixed point, and this implies that the fixed point of loopy BP is a close approximation to the Gibbs distribution. Using these connections we establish that loopy BP quickly converges to the Gibbs distribution when the girth ≥ 6 and λ < λ c (∆).

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Randomly coloring planar graphs with fewer colors than the maximum degree

Cited by 9 publications

References 40 publications

Randomly coloring constant degree graphs

Randomly coloring constant degree graphs

Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

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