Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Galanis, Andreas; Štefankovič, Daniel; Vigoda, Eric

doi:10.1017/s0963548315000401

Cited by 109 publications

(135 citation statements)

References 38 publications

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“…On the other side, Sly [Sly10] proved that, unless NP=RP, it is NP-hard to obtain an FPRAS for Δ-regular graphs in the tree nonuniqueness region (i.e., when λ > λc(Δ)). These results were extended to all 2-spin antiferromagnetic models, for the positive side see [SST12,LLY13] and for the negative side see [SS12,GSV12]. For 2-spin antiferromagnetic models, this establishes a beautiful picture connecting the computational complexity of approximating the partition function to statistical physics phase transitions in the infinite tree.…”

Section: Introductionmentioning

confidence: 71%

Inapproximability for Antiferromagnetic Spin Systems in the Tree Nonuniqueness Region

Galanis

Štefankovič

Vigoda

2015

J. ACM

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A remarkable connection has been established for antiferromagnetic 2-spin systems, including the Ising and hard-core models, showing that the computational complexity of approximating the partition function for graphs with maximum degree Δ undergoes a phase transition that coincides with the statistical physics uniqueness/non-uniqueness phase transition on the infinite Δ-regular tree. Despite this clear picture for 2-spin systems, there is little known for multi-spin systems. We present the first analog of the above inapproximability results for multi-spin systems.The main difficulty in previous inapproximability results was analyzing the behavior of the model on random Δ-regular bipartite graphs, which served as the gadget in the reduction. To this end one needs to understand the moments of the partition function. Our key contribution is connecting: (i) induced matrix norms, (ii) maxima of the expectation of the partition function, and (iii) attractive fixed points of the associated tree recursions (belief propagation). The view through matrix norms allows a simple and generic analysis of the second moment for any spin system on random Δ-regular bipartite graphs. This yields concentration results for any spin system in which one can analyze the maxima of the first moment. The connection to fixed points of the tree recursions enables an analysis of the maxima of the first moment for specific models of interest.For k-colorings we prove that for even k, in the tree nonuniqueness region (specifically for semi-translation invariant measures which corresponds to k < Δ) it is NP-hard, unless NP=RP, to approximate the number of colorings for triangle-free Δ-regular graphs. Our proof extends to the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic model under a mild condition.

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Section: Introductionmentioning

confidence: 71%

Inapproximability for Antiferromagnetic Spin Systems in the Tree Nonuniqueness Region

Galanis

Štefankovič

Vigoda

2015

J. ACM

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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.…”

Section: Path Coupling Distance Functionmentioning

confidence: 94%

“…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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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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“…For general graphical models, if only knowing the parameters B e 's and d, one cannot expect much improvement over this sufficient condition. For example, for the Ising model (ferromagnetic or anti-ferromagnetic, with arbitrary local fields) on graphs with maximum degree ∆ with inverse temperature β, the bound (4) translates to a condition B e = e −2|β| > 1 − 1 4∆ + o( 1 ∆ ); while the sampling problem (even in a static and approximate setting) for the anti-ferromagnetic Ising model in the "non-uniqueness regime", where e −2|β| < 1 − 2 ∆ , is NP-hard [13]. While this sufficient condition on general graphical models focuses on graphical models with soft constraints, our dynamic sampling algorithm is not restricted to such settings.…”

Section: The Running Time Of the Algorithmmentioning

confidence: 99%

“…This bound is asymptotically tight. In the "non-uniqueness regime" where e −2|β| < 1 − 2 ∆ , for the anti-ferromagnetic Ising model, even static and approximate sampling is intractable [13]; and for the ferromagnetic Ising model, by an argument as in [7] there cannot exist such local and parallel sampling algorithms (even for static and approximate sampling) due to the reconstructibility of the ferromagnetic Ising model in the non-uniqueness regime on locally tree-like graphs [3,14].…”

Section: Dynamic Sampling From the Spin Systemsmentioning

confidence: 99%

Dynamic sampling from graphical models

Feng

Vishnoi

Yin

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

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Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Cited by 109 publications

References 38 publications

Inapproximability for Antiferromagnetic Spin Systems in the Tree Nonuniqueness Region

Inapproximability for Antiferromagnetic Spin Systems in the Tree Nonuniqueness Region

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

Dynamic sampling from graphical models

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