A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs

Authors Charlie Carlson , Ewan Davies , Alexandra Kolla, Aditya Potukuchi



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Charlie Carlson
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Ewan Davies
  • Department of Computer Science, Colorado State University, Fort Collins, CO, USA
Alexandra Kolla
  • Computer Science and Engineering, University of California Santa Cruz, CA, USA
Aditya Potukuchi
  • Department of Electrical Engineering and Computer Science, York University, Toronto, Canada

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Charlie Carlson, Ewan Davies, Alexandra Kolla, and Aditya Potukuchi. A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.35

Abstract

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Algorithm design techniques
Keywords
  • approximate counting
  • independent sets
  • bipartite graphs
  • graph containers

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References

  1. J. D. Annan. A Randomised Approximation Algorithm for Counting the Number of Forests in Dense Graphs. Combinatorics, Probability and Computing, 3(3):273-283, 1994. URL: https://doi.org/10.1017/S0963548300001188.
  2. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential Algorithms for Unique Games and Related Problems. Journal of the ACM, 62(5):1-25, 2015. URL: https://doi.org/10.1145/2775105.
  3. Sanjeev Arora, David Karger, and Marek Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 284-293, New York, NY, USA, 1995. Association for Computing Machinery. URL: https://doi.org/10.1145/225058.225140.
  4. Sanjeev Arora, Subhash A. Khot, Alexandra Kolla, David Steurer, Madhur Tulsiani, and Nisheeth K. Vishnoi. Unique games on expanding constraint graphs are easy: Extended abstract. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pages 21-28, Victoria British Columbia Canada, 2008. ACM. URL: https://doi.org/10.1145/1374376.1374380.
  5. Victor Bapst and Amin Coja-Oghlan. Harnessing the Bethe free energy. Random Structures & Algorithms, 49(4):694-741, 2016. URL: https://doi.org/10.1002/rsa.20692.
  6. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding Semidefinite Programming Hierarchies via Global Correlation. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 472-481, 2011. URL: https://doi.org/10.1109/FOCS.2011.95.
  7. Antonio Blanca, Andreas Galanis, Leslie Ann Goldberg, Daniel Stefankovic, Eric Vigoda, and Kuan Yang. Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs. In Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018), volume 116 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1-33:15, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.33.
  8. Sarah Cannon and Will Perkins. Counting independent sets in unbalanced bipartite graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1456-1466, 2020. URL: https://doi.org/10.1137/1.9781611975994.88.
  9. Charles Carlson, Ewan Davies, and Alexandra Kolla. Efficient algorithms for the Potts model on small-set expanders. To appear in Chicago Journal of Theoretical Computer Science, March 2020. URL: https://arxiv.org/abs/2003.01154.
  10. Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, Aditya Potukuchi, and Corrine Yap. Algorithms for the ferromagnetic Potts model on expanders. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 344-355, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00040.
  11. Zongchen Chen, Andreas Galanis, Leslie A. Goldberg, Will Perkins, James Stewart, and Eric Vigoda. Fast algorithms at low temperatures via Markov chains. Random Structures & Algorithms, 58(2):294-321, 2021. URL: https://doi.org/10.1002/rsa.20968.
  12. Fan Chung. Spectral Graph Theory, volume 92 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, December 1996. URL: https://doi.org/10.1090/cbms/092.
  13. Amin Coja-Oghlan, Colin Cooper, and Alan Frieze. An Efficient Sparse Regularity Concept. SIAM Journal on Discrete Mathematics, 23(4):2000-2034, 2010. URL: https://doi.org/10.1137/080730160.
  14. Amin Coja-Oghlan and Will Perkins. Belief Propagation on Replica Symmetric Random Factor Graph Models. In Klaus Jansen, Claire Mathieu, José D. P. Rolim, and Chris Umans, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016), volume 60 of Leibniz International Proceedings in Informatics (LIPIcs), pages 27:1-27:15, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.27.
  15. Amin Coja-Oghlan and Will Perkins. Bethe States of Random Factor Graphs. Communications in Mathematical Physics, 366(1):173-201, February 2019. URL: https://doi.org/10.1007/s00220-019-03387-7.
  16. Martin Dyer, Alan Frieze, and Mark Jerrum. Approximately counting Hamilton cycles in dense graphs. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '94, pages 336-343, USA, 1994. Society for Industrial and Applied Mathematics. URL: https://dl.acm.org/doi/abs/10.5555/314464.314557.
  17. Martin Dyer, Leslie Ann Goldberg, Catherine Greenhill, and Mark Jerrum. The Relative Complexity of Approximate Counting Problems. Algorithmica, 38(3):471-500, 2004. URL: https://doi.org/10.1007/s00453-003-1073-y.
  18. A. Frieze. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, FOCS '96, page 21, USA, 1996. IEEE Computer Society. Google Scholar
  19. A. Frieze and R. Kannan. The regularity lemma and approximation schemes for dense problems. In Proceedings of 37th Conference on Foundations of Computer Science, pages 12-20, 1996. URL: https://doi.org/10.1109/SFCS.1996.548459.
  20. Andreas Galanis, Qi Ge, Daniel Štefankovič, Eric Vigoda, and Linji Yang. Improved inapproximability results for counting independent sets in the hard-core model. Random Structures & Algorithms, 45(1):78-110, 2014. URL: https://doi.org/10.1002/rsa.20479.
  21. Andreas Galanis, Leslie Ann Goldberg, and James Stewart. Fast Algorithms for General Spin Systems on Bipartite Expanders. ACM Transactions on Computation Theory, 13(4):25:1-25:18, 2021. URL: https://doi.org/10.1145/3470865.
  22. David Galvin. A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube. Combinatorics, Probability and Computing, 20(1):27-51, 2011. URL: https://doi.org/10.1017/S0963548310000155.
  23. David Galvin and Prasad Tetali. Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs. Random Structures & Algorithms, 28(4):427-443, 2006. URL: https://doi.org/10.1002/rsa.20094.
  24. Leslie Ann Goldberg, John Lapinskas, and David Richerby. Faster exponential-time algorithms for approximately counting independent sets. Theoretical Computer Science, 892:48-84, November 2021. URL: https://doi.org/10.1016/j.tcs.2021.09.009.
  25. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 482-491. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.36.
  26. Tyler Helmuth, Matthew Jenssen, and Will Perkins. Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 59(2):817-848, 2023. URL: https://doi.org/10.1214/22-AIHP1263.
  27. Tyler Helmuth, Will Perkins, and Guus Regts. Algorithmic PirogovendashSinai theory. Probability Theory and Related Fields, 2019. URL: https://doi.org/10.1007/s00440-019-00928-y.
  28. Vishesh Jain, Frederic Koehler, and Andrej Risteski. Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: A unified perspective. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 1226-1236, Phoenix, AZ, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316299.
  29. Matthew Jenssen, Peter Keevash, and Will Perkins. Algorithms for #BIS-Hard Problems on Expander Graphs. SIAM Journal on Computing, 49(4):681-710, 2020. URL: https://doi.org/10.1137/19M1286669.
  30. Matthew Jenssen and Will Perkins. Independent sets in the hypercube revisited. Journal of the London Mathematical Society, 102(2):645-669, 2020. URL: https://doi.org/10.1112/jlms.12331.
  31. Matthew Jenssen, Will Perkins, and Aditya Potukuchi. Independent sets of a given size and structure in the hypercube. Combinatorics, Probability and Computing, 31(4):702-720, July 2022. URL: https://doi.org/10.1017/S0963548321000559.
  32. Matthew Jenssen, Will Perkins, and Aditya Potukuchi. Approximately counting independent sets in bipartite graphs via graph containers. Random Structures & Algorithms, 63(1):215-241, 2023. URL: https://doi.org/10.1002/rsa.21145.
  33. Mark Jerrum and Alistair Sinclair. Approximating the Permanent. SIAM Journal on Computing, 18(6):1149-1178, 1989. URL: https://doi.org/10.1137/0218077.
  34. Jeff Kahn and Jinyoung Park. The Number of Maximal Independent Sets in the Hamming Cube. Combinatorica, 42(6):853-880, December 2022. URL: https://doi.org/10.1007/s00493-021-4729-9.
  35. Frederic Koehler, Holden Lee, and Andrej Risteski. Sampling Approximately Low-Rank Ising Models: MCMC meets Variational Methods. In Proceedings of Thirty Fifth Conference on Learning Theory, pages 4945-4988. PMLR, June 2022. URL: https://proceedings.mlr.press/v178/koehler22a.html.
  36. Alexandra Kolla. Spectral Algorithms for Unique Games. In 2010 IEEE 25th Annual Conference on Computational Complexity, pages 122-130, Cambridge, MA, USA, June 2010. IEEE. URL: https://doi.org/10.1109/CCC.2010.20.
  37. Alexandra Kolla and Madhur Tulsiani. Playing random and expanding unique games. Unpublished, 2007. URL: https://home.cs.colorado.edu/~alko5368/UGspec.pdf.
  38. Jingcheng Liu and Pinyan Lu. FPTAS for #BIS with Degree Bounds on One Side. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 549-556, New York, NY, USA, June 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746598.
  39. Shayan Oveis Gharan and Luca Trevisan. A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs. Theory of Computing, 11(1):241-256, 2015. URL: https://doi.org/10.4086/toc.2015.v011a009.
  40. Jinyoung Park. Note on the Number of Balanced Independent Sets in the Hamming Cube. The Electronic Journal of Combinatorics, page P2.34, 2022. URL: https://doi.org/10.37236/10471.
  41. J. Scott Provan and Michael O. Ball. The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM Journal on Computing, 12(4):777-788, 1983. URL: https://doi.org/10.1137/0212053.
  42. Andrej Risteski. How to calculate partition functions using convex programming hierarchies: Provable bounds for variational methods. In Conference on Learning Theory, pages 1402-1416. PMLR, 2016. URL: https://proceedings.mlr.press/v49/risteski16.html.
  43. A. A. Sapozhenko. On the number of connected subsets with given cardinality of the boundary in bipartite graphs. Metody Diskretnogo Analiza, (45):42-70, 96, 1987. Google Scholar
  44. A. A. Sapozhenko. On the number of independent sets in extenders. Diskretnaya Matematika, 13(1):56-62, 2001. URL: https://doi.org/10.1515/dma.2001.11.2.155.
  45. A. Sly and N. Sun. The Computational Hardness of Counting in Two-Spin Models on d-Regular Graphs. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 361-369, 2012. URL: https://doi.org/10.1109/FOCS.2012.56.
  46. Allan Sly. Computational Transition at the Uniqueness Threshold. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 287-296, Las Vegas, NV, USA, 2010. IEEE. URL: https://doi.org/10.1109/FOCS.2010.34.