Improved inapproximability results for counting independent sets in the hard‐core model

Galanis, Andreas; Ge, Qi; Štefankovič, Daniel; Vigoda, Eric; Yang, Linji

doi:10.1002/rsa.20479

Cited by 27 publications

(8 citation statements)

References 19 publications

(55 reference statements)

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“…We will prove here that the convergence of BP provides the existence of a distance function Φ satisfying (7). We defer the technical proof of (6) to Section A of the appendix. The Jacobian J of the BP operator F is given by…”

Section: Path Coupling Distance Functionmentioning

confidence: 95%

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

Efthymiou¹,

Hayes

Štefankovič

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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Section: Path Coupling Distance Functionmentioning

confidence: 95%

“…On the other side, Sly [33] (extended in [6,7,34,8]) has established that, unless N P = RP , for all ∆ ≥ 3, there exists γ > 0, for all λ > λ c (∆), there is no polynomial-time algorithm for triangle-free ∆-regular graphs to approximate the partition function within a factor 2 γn .…”

Section: Introduction Backgroundmentioning

confidence: 99%

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

Efthymiou¹,

Hayes

Štefankovič

et al. 2016

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

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“…Note that all known approximation algorithms for the partition function of the uniform hard-core model, apply to instances (𝐺, 𝜆), where 𝐺 has maximum degree 𝛥 and 𝜆 < 𝜆 c (𝛥) = (𝛥−1) 𝛥−1 (𝛥−2) 𝛥 ∼ e 𝛥 . This parameter regime is tight, as approximating the partition function for a hard-core model is known to be NP-hard for instances (𝐺, 𝜆) with 𝜆 > 𝜆 c (𝛥) [20,37]. We obtain our algorithms by bounding the maximum degree of 𝐺 𝑋 and using the known approximation algorithms for the hard-core model.…”

Section: Approximation Algorithms Via Canonical Discretizationmentioning

confidence: 99%

Algorithms for hard-constraint point processes via discretization

Friedrich¹,

Göbel²,

Katzmann³

et al. 2021

Preprint

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We study a general model for continuous spin systems with hard-core interactions. Our model allows for a mixture of 𝑞 types of particles on a 𝑑-dimensional Euclidean region V of volume 𝜈 (V). For each type, particle positions are distributed according to a Poisson point process. The Gibbs distribution over all possible system states is characterized by the mixture of these point processes conditioned that no two particles are closer than some distance parameterized by a 𝑞 ×𝑞 matrix. This general model encompasses classical continuous spin systems, such as the hard-sphere model or the Widom-Rowlinson model.We present su cient conditions for approximating the partition function of this model, which is the normalizing factor of its Gibbs distribution. For the hard-sphere model, our method yields a randomized approximation algorithm with running time polynomial in 𝜈 (V) for the known uniqueness regime of the Gibbs measure. In the same parameter regime, we obtain a quasi-polynomial deterministic approximation algorithm for the hard-sphere model, which, to our knowledge, is the rst rigorous deterministic approximation algorithm for a continuous spin system. We obtain similar approximation results for all continuous spin systems captured by our model and, in particular, the rst explicit approximation bounds for the Widom-Rowlinson model. Additionally, we show how to obtain e cient approximate samplers for the Gibbs distributions of the respective spin systems within the same parameter regimes.Key to our method is reducing the continuous model to a discrete instance of the hard-core model with size polynomial in 𝜈 (V). This generalizes existing discretization schemes for the hard-sphere model and, additionally, improves the required number of vertices of the generated graph from super-exponential to quadratic in 𝜈 (V), which we argue to be tight.

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“…this line of work culminated in papers by Dyer and Greenhill [5] and Vigoda [29], who gave MCMC based FPRASs for λ < 2/(d − 1) for graphs of maximum degree at most d + 1. Weitz [31] (see also [2]) introduced a new paradigm by using correlation decay directly to design a deterministic FPTAS and gave an algorithm under the condition λ < λ c (d) for graphs of degree at most d + 1; this range of applicability was later proved to be optimal by Sly [26] (see also [7,27]). To date, no MCMC based algorithm is known to have a range of applicability as wide as Weitz's algorithm.…”

Section: Techniquesmentioning

confidence: 99%

“…(Note that the condition on λ is only in terms of the d-ary tree, while the FPTAS applies to all graphs.) This connection to phase transitions was further strengthened by Sly [26] (see also [7,27]), who showed that an FPRAS for the partition function of the hard core model with λ > λ c (d) on graphs of degree d + 1 would imply NP = RP.…”

mentioning

confidence: 92%

Spatial mixing and the connective constant: Optimal bounds

Sinclair

Srivastava

Štefankovič

et al. 2014

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

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We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model (which is defined as a weighted sum over all matchings where each matching is given a weight γ |V |−2|M | in terms of a fixed parameter γ called the monomer activity) and the hard core model (which is defined as a weighted sum over all independent sets where an independent set I is given a weight λ |I| in terms of a fixed parameter λ called the vertex activity). The connective constant is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős-Rényi model G(n, d/n).Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called message approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain FPTASs for the two problems on graphs of bounded connective constant.In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati, Gamarnik, Katz, Nair and Tetali [STOC 2007], and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant Δ whenever

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Improved inapproximability results for counting independent sets in the hard‐core model

Cited by 27 publications

References 19 publications

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

Algorithms for hard-constraint point processes via discretization

Spatial mixing and the connective constant: Optimal bounds

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