On the hardness of sampling independent sets beyond the tree threshold

Mossel, Elchanan; Weitz, Dror; Wormald, Nicholas C.

doi:10.1007/s00440-007-0131-9

Cited by 82 publications

(172 citation statements)

References 30 publications

(92 reference statements)

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“…Since c 2 = C 0 ∩ V c and c 3 = C 1 ∩ V a , condition (19) requires there are no white particles in C 0 and no gray particles in C 1 . Figure 12 shows a hard-core configuration that satisfies this initial condition.…”

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

“…Example of hard-core configuration on the 6 × 9 triangular grid that satisfies (19) [Color figure can be viewed at wileyonlinelibrary.com] 13 The configurations 1 and 2 corresponding to the initial configuration in Figure 12 We now describe in detail how to construct each of the paths (j) for j = 1, … , 2L. We distinguish 2 cases, depending on whether (a) j has a vertical bridge in column c 3j+2 or (b) not, see the 2 examples in Figure 13.…”

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

“…The second path (2) ∶ * → b is then constructed by means of the energy reduction algorithm by columns, which can be used since the configuration * satisfies condition (19) and hence is a suitable initial configuration for the algorithm. Since configuration * has no vertical c-bridges (see case (b) for the energy reduction algorithm by columns), the procedure guarantees that For case (b), consider the configuration * that differs from a only in the sites of the first horizontal stripe S 0 , namely * (v) ∶=…”

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

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Tunneling of the hard‐core model on finite triangular lattices

Zocca

2018

Random Struct Algorithms

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We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.

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Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

Section: Energy Reduction Algorithm By Columnsmentioning

confidence: 99%

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Tunneling of the hard‐core model on finite triangular lattices

Zocca

2018

Random Struct Algorithms

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“…In particular, the question is if the threshold can be formulated in terms of the coloring model on a branching process tree with Poisson(d) degree distribution. One would expect rapid mixing for in the "uniqueness phase", but perhaps even beyond it, see [11,21,22]. …”

Section: Related Workmentioning

confidence: 99%

Gibbs rapidly samples colorings of G(n, d/n)

Mossel

Sly

2009

Probab. Theory Relat. Fields

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Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erdős-Rényi random graph G (n, d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in, it contains many nodes of degree of order (log n)/(log log n). The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n, d/n) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n 1+ (1/ log log n) with high probability. High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. Almost all known sufficient conditions in terms of number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work we consider sampling q-colorings and show that for every d < ∞ there exists q(d) < ∞ such that for all q ≥ q(d) the mixing time of the Gibbs sampling on G(n, d/n) is polynomial in n with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n, d/n) for d > 1 where the number of colors does not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). In previous work we have shown that similar results hold for the ferromagnetic Ising model. However, the proof for the Ising model crucially relied on monotonicity arguments and the "Weitz tree", both of which have no counterparts in the coloring setting. Our proof presented here exploits in novel ways the local treelike structure of Erdős-Rényi random graphs, block dynamics, spatial decay properties and coupling arguments. Our results give the first polynomial-time algorithm to approximately sample colorings on G(n, d/n) with a constant number of colors. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which there exists an α > 0 such that every vertex v of the graph has a neighborhood N (v) of radius O(log n) in which the induced sub-graph is the union of a tree and at most O(1) edges and where each simple path of length O(log n) satisfies u∈ v =u α d (u,v) = O(log n). The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures.

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“…To establish the lower bound we introduce a construction reminiscent of those used in . Namely, starting from the sequence

\underset{true_}{scriptM}

(Δ, Ω, Ψ, Θ)

‐models, we define another sequence

{\underset{true_}{scriptM}}^{\otimes} = {({scriptM}_{n}^{\otimes})}_{n}

of models as follows.…”

Section: The Bethe Free Energymentioning

confidence: 99%

Harnessing the Bethe free energy

Bapst

Coja-Oghlan

2016

Random Struct Algorithms

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A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the k‐SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the “replica symmetric cavity method” yields the correct value of the partition function if the underlying model enjoys certain properties [Krzkala et al., PNAS (2007) 10318–10323]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a “regularity lemma” for probability measures on sets of the form Ωn for a finite Ω and a large n that may be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 694–741, 2016

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On the hardness of sampling independent sets beyond the tree threshold

Cited by 82 publications

References 30 publications

Tunneling of the hard‐core model on finite triangular lattices

Tunneling of the hard‐core model on finite triangular lattices

Gibbs rapidly samples colorings of G(n, d/n)

Harnessing the Bethe free energy

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