research-article

Fast and Perfect Sampling of Subgraphs and Polymer Systems

Authors:

Antonio Blanca,

Will PerkinsAuthors Info & Claims

ACM Transactions on Algorithms, Volume 20, Issue 1

Article No.: 5, Pages 1 - 30

https://doi.org/10.1145/3632294

Published: 22 January 2024 Publication History

Abstract

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

References

[1]

Matteo Agostini, Marco Bressan, and Shahrzad Haddadan. 2019. Mixing time bounds for graphlet random walks. Inform. Process. Lett. 152 (2019), 105851.

[2]

Konrad Anand, Andreas Göbel, Marcus Pappik, and Will Perkins. 2023. Perfect sampling for hard spheres from strong spatial mixing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM’23), Vol. 275. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, 38:1–38:18.

[3]

Konrad Anand and Mark Jerrum. 2022. Perfect sampling in infinite spin systems via strong spatial mixing. SIAM J. Comput. 51, 4 (2022), 1280–1295.

[4]

Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. 2021. Spectral independence in high-dimensional expanders and applications to the hardcore model. SIAM J. Comput.0 (2021), FOCS20–1.

Digital Library

[5]

Kim Baskerville, Peter Grassberger, and Maya Paczuski. 2007. Graph animals, subgraph sampling, and motif search in large networks. Physical Review E 76, 3 (2007), 036107.

[6]

Mansurul A. Bhuiyan, Mahmudur Rahman, Mahmuda Rahman, and Mohammad Al Hasan. 2012. Guise: Uniform sampling of graphlets for large graph analysis. In Proceedings of the 2012 IEEE 12th International Conference on Data Mining. IEEE, 91–100.

Digital Library

[7]

Christian Borgs, Jennifer Chayes, Tyler Helmuth, Will Perkins, and Prasad Tetali. 2020. Efficient sampling and counting algorithms for the Potts model on \(\mathbb {Z}^d\) at all temperatures. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC’20). 738–751.

Digital Library

[8]

Christian Borgs, Jennifer Chayes, Jeff Kahn, and László Lovász. 2013. Left and right convergence of graphs with bounded degree. Random Structures & Algorithms 42, 1 (2013), 1–28.

Digital Library

[9]

Christian Borgs and John Z. Imbrie. 1989. A unified approach to phase diagrams in field theory and statistical mechanics. Communications in Mathematical Physics 123, 2 (1989), 305–328.

[10]

Marco Bressan. 2021. Efficient and near-optimal algorithms for sampling connected subgraphs. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC’21). 1132–1143.

Digital Library

[11]

Marco Bressan, Flavio Chierichetti, Ravi Kumar, Stefano Leucci, and Alessandro Panconesi. 2017. Counting graphlets: Space vs time. In Proceedings of the 10th ACM International Conference on Web Search and Data Mining (WSDM’17). 557–566.

Digital Library

[12]

Marco Bressan, Flavio Chierichetti, Ravi Kumar, Stefano Leucci, and Alessandro Panconesi. 2018. Motif counting beyond five nodes. ACM Transactions on Knowledge Discovery from Data (TKDD) 12, 4 (2018), 1–25.

Digital Library

[13]

Marco Bressan, Stefano Leucci, and Alessandro Panconesi. 2021. Faster motif counting via succinct color coding and adaptive sampling. ACM Transactions on Knowledge Discovery from Data (TKDD) 15, 6 (2021), 1–27.

Digital Library

[14]

Sarah Cannon and Will Perkins. 2020. Counting independent sets in unbalanced bipartite graphs. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). 1456–1466.

Digital Library

[15]

Charles Carlson, Ewan Davies, and Alexandra Kolla. 2020. Efficient algorithms for the Potts model on small-set expanders. arXiv:2003.01154. Retrieved from https://arxiv.org/abs/2003.01154

[16]

Xiaowei Chen, Yongkun Li, Pinghui Wang, and John C. S. Lui. 2016. A general framework for estimating graphlet statistics via random walk. Proceedings of the VLDB Endowment 10, 3 (2016), 253–264.

Digital Library

[17]

Zongchen Chen, Andreas Galanis, Leslie A Goldberg, Will Perkins, James Stewart, and Eric Vigoda. 2021. Fast algorithms at low temperatures via Markov chains. Random Structures & Algorithms 58, 2 (2021), 294–321.

[18]

Zongchen Chen, Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. 2022. Sampling colorings and independent sets of random regular bipartite graphs in the non-uniqueness region. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’22). SIAM, 2198–2207.

[19]

Zongchen Chen, Kuikui Liu, and Eric Vigoda. 2020. Rapid mixing of Glauber dynamics up to uniqueness via contraction. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS’20). IEEE, 1307–1318.

[20]

Zongchen Chen, Kuikui Liu, and Eric Vigoda. 2021. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC’21). 1537–1550.

Digital Library

[21]

Matthew Coulson, Ewan Davies, Alexandra Kolla, Viresh Patel, and Guus Regts. 2020. Statistical physics approaches to unique games. In Proceedings of the 35th Computational Complexity Conference (CCC’20).

Digital Library

[22]

Martin Dyer, Leslie Ann Goldberg, Catherine Greenhill, and Mark Jerrum. 2004. The relative complexity of approximate counting problems. Algorithmica 38, 3 (2004), 471–500.

[23]

Sacha Friedli and Yvan Velenik. 2017. Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press.

[24]

Tobias Friedrich, Andreas Göbel, Martin S Krejca, and Marcus Pappik. 2020. Polymer dynamics via cliques: New conditions for approximations. arXiv:2007.08293. Retrieved from https://arxiv.org/abs/2007.08293

[25]

Andreas Galanis, Leslie Ann Goldberg, and James Stewart. 2021. Fast algorithms for general spin systems on bipartite expanders. ACM Transactions on Computation Theory (TOCT) 13, 4 (2021), 1–18.

Digital Library

[26]

Andreas Galanis, Leslie Ann Goldberg, and James Stewart. 2022. Fast mixing via polymers for random graphs with unbounded degree. Information and Computation 285, Part B (2022), 104894.

Digital Library

[27]

Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. 2016. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Combinatorics, Probability and Computing 25, 4 (2016), 500–559.

[28]

Oded Goldreich and Dana Ron. 1997. Property testing in bounded degree graphs. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC’97). 406–415.

Digital Library

[29]

Joshua A. Grochow and Manolis Kellis. 2007. Network motif discovery using subgraph enumeration and symmetry-breaking. In Annual International Conference on Research in Computational Molecular Biology. Springer, 92–106.

[30]

Christian Gruber and Hervé Kunz. 1971. General properties of polymer systems. Communications in Mathematical Physics 22, 2 (1971), 133–161.

[31]

Olle Haggstrom and Karin Nelander. 1999. On exact simulation of Markov random fields using coupling from the past. Scandinavian Journal of Statistics 26, 3 (1999), 395–411.

[32]

Kun He, Chunyang Wang, and Yitong Yin. 2022. Sampling Lovász local lemma for general constraint satisfaction solutions in near-linear time. In Proceedings of the 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS’22). IEEE, 147–158.

[33]

Kun He, Kewen Wu, and Kuan Yang. 2023. Improved bounds for sampling solutions of random CNF formulas. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’23). SIAM, 3330–3361.

[34]

Tyler Helmuth, Matthew Jenssen, and Will Perkins. 2023. Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs. In Annales de l’Institut Henri Poincare (B) Probabilites et statistiques, Vol. 59. Institut Henri Poincaré, 817–848.

[35]

Tyler Helmuth, Will Perkins, and Guus Regts. 2020. Algorithmic Pirogov–Sinai theory. Probability Theory and Related Fields 176, 3 (2020), 851–895.

[36]

Mark Huber. 2004. Perfect sampling using bounding chains. The Annals of Applied Probability 14, 2 (2004), 734–753.

[37]

Svante Janson, Andrzej Rucinski, and Tomasz Luczak. 2011. Random Graphs. John Wiley & Sons.

[38]

Matthew Jenssen, Peter Keevash, and Will Perkins. 2020. Algorithms for #BIS-hard problems on expander graphs. SIAM J. Comput. 49, 4 (2020), 681–710.

Digital Library

[39]

Matthew Jenssen, Aditya Potukuchi, and Will Perkins. 2022. Approximately counting independent sets in bipartite graphs via graph containers. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’22). SIAM, 499–516.

[40]

Madhav Jha, C. Seshadhri, and Ali Pinar. 2015. Path sampling: A fast and provable method for estimating 4-vertex subgraph counts. In Proceedings of the 24th International Conference on World Wide Web (WWW’15). 495–505.

Digital Library

[41]

Reza Kahkeshani. 2013. A generalization of the Catalan numbers. Journal of Integer Sequences 16, 2 (2013), 3.

[42]

Nadav Kashtan, Shalev Itzkovitz, Ron Milo, and Uri Alon. 2004. Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs. Bioinformatics 20, 11 (2004), 1746–1758.

Digital Library

[43]

Roman Koteckỳ and David Preiss. 1986. Cluster expansion for abstract polymer models. Communications in Mathematical Physics 103, 3 (1986), 491–498.

[44]

Timo Kötzing and Martin S. Krejca. 2019. First-hitting times under drift. Theoretical Computer Science 796 (2019), 51–69.

Digital Library

[45]

Chao Liao, Jiabao Lin, Pinyan Lu, and Zhenyu Mao. 2019. Counting independent sets and colorings on random regular bipartite graphs. In Proceedings of the Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM’19). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

[46]

Xuesong Lu and Stéphane Bressan. 2012. Sampling connected induced subgraphs uniformly at random. In Proceedings of the International Conference on Scientific and Statistical Database Management. Springer, 195–212.

Digital Library

[47]

Ryuta Matsuno and Aristides Gionis. 2020. Improved mixing time for k-subgraph sampling. In Proceedings of the 2020 SIAM International Conference on Data Mining. SIAM, 568–576.

[48]

Kirill Paramonov, Dmitry Shemetov, and James Sharpnack. 2019. Estimating graphlet statistics via lifting. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. 587–595.

Digital Library

[49]

James Gary Propp and David Bruce Wilson. 1996. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures & Algorithms 9, 1-2 (1996), 223–252.

Digital Library

[50]

Andrew Read-McFarland and Daniel Štefankovič. 2021. The hardness of sampling connected subgraphs. In Proceedings of the Latin American Symposium on Theoretical Informatics. Springer, 464–475.

[51]

Nino Shervashidze, SVN Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten Borgwardt. 2009. Efficient graphlet kernels for large graph comparison. In Proceedings of the Artificial Intelligence and Statistics. PMLR, 488–495.

[52]

Allan Sly. 2010. Computational transition at the uniqueness threshold. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science (FOCS’10). IEEE, 287–296.

Digital Library

[53]

Allan Sly and Nike Sun. 2012. The computational hardness of counting in two-spin models on d-regular graphs. In Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science. IEEE, 361–369.

Digital Library

[54]

Dror Weitz. 2006. Counting independent sets up to the tree threshold. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing. 140–149.

Digital Library

[55]

Nacu Şerban and Peres Yuval. 2005. Fast simulation of new coins from old. The Annals of Applied Probability 15, 1A (2005), 93–115.

Cited By

Blanca AGheissari R(2023)Sampling from the Potts model at low temperatures via Swendsen–Wang dynamics2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00122(2006-2020)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00122
Chen XLiu JYin Y(2023)Uniqueness and Rapid Mixing in the Bipartite Hardcore Model (extended abstract)2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00121(1991-2005)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00121
Carlson CDavies EFraiman NKolla APotukuchi AYap C(2022)Algorithms for the ferromagnetic Potts model on expanders2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00040(344-355)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00040

Index Terms

Fast and Perfect Sampling of Subgraphs and Polymer Systems
1. Mathematics of computing
  1. Probability and statistics
    1. Probabilistic algorithms
    2. Stochastic processes
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Generating random combinatorial structures
    2. Random walks and Markov chains

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 20, Issue 1

January 2024

297 pages

EISSN:1549-6333

DOI:10.1145/3613497

Editor:
Edith Cohen
Google Research, USA and Tel Aviv University, Israel

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Publication History

Published: 22 January 2024

Online AM: 10 November 2023

Accepted: 23 October 2023

Revised: 15 May 2023

Received: 01 June 2022

Published in TALG Volume 20, Issue 1

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Cited By

Blanca AGheissari R(2023)Sampling from the Potts model at low temperatures via Swendsen–Wang dynamics2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00122(2006-2020)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00122
Chen XLiu JYin Y(2023)Uniqueness and Rapid Mixing in the Bipartite Hardcore Model (extended abstract)2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00121(1991-2005)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00121
Carlson CDavies EFraiman NKolla APotukuchi AYap C(2022)Algorithms for the ferromagnetic Potts model on expanders2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00040(344-355)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00040

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