Sampling in uniqueness from the potts and random-cluster models on random regular graphs

Antonio Blanca Pimentel, Andreas Galanis, Leslie Ann Goldberg, Daniel Štefankovic, Eric Vigoda, Kuan Yang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q ≥ 3 and Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Δ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p, q on random Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018
EditorsEric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770859
DOIs
StatePublished - Aug 1 2018
Event21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018 - Princeton, United States
Duration: Aug 20 2018Aug 22 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume116
ISSN (Print)1868-8969

Conference

Conference21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018
CountryUnited States
CityPrinceton
Period8/20/188/22/18

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Potts model
Sampling
Temperature

All Science Journal Classification (ASJC) codes

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Cite this

Pimentel, A. B., Galanis, A., Goldberg, L. A., Štefankovic, D., Vigoda, E., & Yang, K. (2018). Sampling in uniqueness from the potts and random-cluster models on random regular graphs. In E. Blais, J. D. P. Rolim, D. Steurer, & K. Jansen (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018 [33] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.33
Pimentel, Antonio Blanca ; Galanis, Andreas ; Goldberg, Leslie Ann ; Štefankovic, Daniel ; Vigoda, Eric ; Yang, Kuan. / Sampling in uniqueness from the potts and random-cluster models on random regular graphs. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. editor / Eric Blais ; Jose D. P. Rolim ; David Steurer ; Klaus Jansen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. (Leibniz International Proceedings in Informatics, LIPIcs).
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abstract = "We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q ≥ 3 and Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Δ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p, q on random Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.",
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Pimentel, AB, Galanis, A, Goldberg, LA, Štefankovic, D, Vigoda, E & Yang, K 2018, Sampling in uniqueness from the potts and random-cluster models on random regular graphs. in E Blais, JDP Rolim, D Steurer & K Jansen (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018., 33, Leibniz International Proceedings in Informatics, LIPIcs, vol. 116, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation, RANDOM 2018, Princeton, United States, 8/20/18. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.33

Sampling in uniqueness from the potts and random-cluster models on random regular graphs. / Pimentel, Antonio Blanca; Galanis, Andreas; Goldberg, Leslie Ann; Štefankovic, Daniel; Vigoda, Eric; Yang, Kuan.

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. ed. / Eric Blais; Jose D. P. Rolim; David Steurer; Klaus Jansen. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. 33 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 116).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AU - Pimentel, Antonio Blanca

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AU - Štefankovic, Daniel

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N2 - We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q ≥ 3 and Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Δ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p, q on random Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.

AB - We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q ≥ 3 and Δ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Δ-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Δ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p, q on random Δ-regular graphs for all values of q 1 and p < pc(q, Δ), where pc(q, Δ) corresponds to a uniqueness threshold for the model on the Δ-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Δ-regular graphs.

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Pimentel AB, Galanis A, Goldberg LA, Štefankovic D, Vigoda E, Yang K. Sampling in uniqueness from the potts and random-cluster models on random regular graphs. In Blais E, Rolim JDP, Steurer D, Jansen K, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2018. 33. (Leibniz International Proceedings in Informatics, LIPIcs). https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.33