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On the Spectral Expansion of Monotone Subsets of the Hypercube

Authors Yumou Fei , Renato Ferreira Pinto Jr.

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov processes
  • Theory of computation → Random walks and Markov chains
Keywords
  • Random walks
  • mixing time
  • FKG inequality
  • Poincaré inequality
  • directed isoperimetry

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Abstract

We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset A ⊆ {0,1}ⁿ of density μ(A), the previous best lower bound on the spectral gap, due to Cohen [Cohen, 2016], was γ ≳ μ(A)/n², improving upon the earlier bound γ ≳ μ(A)²/n² established by Ding and Mossel [Ding and Mossel, 2014]. In this paper, we prove the optimal lower bound γ ≳ μ(A)/n. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from O(n³), as shown by Ding and Mossel, to O(n²). Along the way, we develop two new inequalities that may be of independent interest: (1) a directed L²-Poincaré inequality on the hypercube, and (2) an "approximate" FKG inequality for monotone sets.

Cite As Get BibTex

Yumou Fei and Renato Ferreira Pinto Jr.. On the Spectral Expansion of Monotone Subsets of the Hypercube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 42:1-42:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.42

Author Details

Yumou Fei
  • Department of EECS, Massachusetts Institute of Technology, Cambridge, MA, USA
Renato Ferreira Pinto Jr.
  • Cheriton School of Computer Science, University of Waterloo, Canada

Funding

  • Ferreira Pinto Jr., Renato: Supported by an NSERC Canada Graduate Scholarship.

Acknowledgements

We thank Lap Chi Lau for many helpful insights and discussions on spectral theory for directed graphs, and Esty Kelman for early discussions in the development of this project.

References

  1. Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 1198-1211, 2020. URL: https://doi.org/10.1145/3357713.3384317.
  2. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials ii: high-dimensional walks and an fpras for counting bases of a matroid. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 1-12, 2019. URL: https://doi.org/10.1145/3313276.3316385.
  3. Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and Geometry of Markov Diffusion Operators. Springer Cham, 2014. Google Scholar
  4. Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 164-173, 2014. URL: https://doi.org/10.1145/2591796.2591887.
  5. Hadley Black, Deeparnab Chakrabarty, and C Seshadhri. A d^1/2 + o(1) monotonicity tester for boolean functions on d-dimensional hypergrids. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 1796-1821, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00110.
  6. Hadley Black, Deeparnab Chakrabarty, and C Seshadhri. Directed isoperimetric theorems for boolean functions on the hypergrid and an O(n √d) monotonicity tester. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 233-241, 2023. Google Scholar
  7. Hadley Black, Deeparnab Chakrabarty, and Comandur Seshadhri. A o(d) ⋅ polylog n monotonicity tester for boolean functions over the hypergrid [n]^d. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2133-2151, 2018. Google Scholar
  8. Hadley Black, Iden Kalemaj, and Sofya Raskhodnikova. Isoperimetric inequalities for real-valued functions with applications to monotonicity testing. Random Structures & Algorithms, 2024. Google Scholar
  9. Vladimir Igorevich Bogachev. Measure theory, volume 1. Springer, 2007. Google Scholar
  10. Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer. Improved monotonicity testers via hypercube embeddings. In Proceedings of the Innovations in Theoretical Computer Science Conference (ITCS), 2023. Google Scholar
  11. D Chakrabarty and C Seshadhri. An o(n) monotonicity tester for boolean functions over the hypercube. SIAM Journal on Computing, 45(2):461-472, 2016. URL: https://doi.org/10.1137/13092770X.
  12. Deeparnab Chakrabarty, Xi Chen, Simeon Ristic, C Seshadhri, and Erik Waingarten. Monotonicity testing of high-dimensional distributions with subcube conditioning, 2025. URL: https://doi.org/10.48550/arXiv.2502.16355.
  13. Fan Chung. Laplacians and the cheeger inequality for directed graphs. Annals of Combinatorics, 9:1-19, 2005. Google Scholar
  14. Emma Cohen. Problems in catalan mixing and matchings in regular hypergraphs. Georgia Tech PhD thesis, 2016. Google Scholar
  15. Mary Cryan, Heng Guo, and Giorgos Mousa. Modified log-sobolev inequalities for strongly log-concave distributions. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 1358-1370, 2019. URL: https://doi.org/10.1109/FOCS.2019.00083.
  16. Jian Ding and Elchanan Mossel. Mixing under monotone censoring. Electronic Communications in Probability, 19:1-6, 2014. Google Scholar
  17. Martin Dyer. Approximate counting by dynamic programming. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 693-699, 2003. URL: https://doi.org/10.1145/780542.780643.
  18. Weiming Feng and Ce Jin. Approximately counting knapsack solutions in subquadratic time. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1094-1135, 2025. URL: https://doi.org/10.1137/1.9781611978322.32.
  19. Renato Ferreira Pinto Jr. Directed poincaré inequalities and L¹ monotonicity testing of lipschitz functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), 2023. URL: https://doi.org/LIPIcs.APPROX/RANDOM.2023.61.
  20. Renato Ferreira Pinto Jr. Directed isoperimetry and monotonicity testing: A dynamical approach. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 2295-2305. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00134.
  21. James Allen Fill. Eigenvalue bounds on convergence to stationarity for nonreversible markov chains, with an application to the exclusion process. The annals of applied probability, pages 62-87, 1991. Google Scholar
  22. Cees M Fortuin, Pieter W Kasteleyn, and Jean Ginibre. Correlation inequalities on some partially ordered sets. Communications in Mathematical Physics, 22:89-103, 1971. Google Scholar
  23. Kaito Fujii, Tasuku Soma, and Yuichi Yoshida. Polynomial-time algorithms for submodular laplacian systems. Theoretical Computer Science, 892:170-186, 2021. URL: https://doi.org/10.1016/J.TCS.2021.09.019.
  24. Pawel Gawrychowski, Liran Markin, and Oren Weimann. A faster FPTAS for #knapsack. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 64-1, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.64.
  25. Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 3(20):301-337, 2000. URL: https://doi.org/10.1007/S004930070011.
  26. Parikshit Gopalan, Adam Klivans, Raghu Meka, Daniel Štefankovic, Santosh Vempala, and Eric Vigoda. An fptas for #knapsack and related counting problems. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 817-826, 2011. URL: https://doi.org/10.1109/FOCS.2011.32.
  27. Venkatesan Guruswami. Rapidly mixing markov chains: A comparison of techniques (a survey), 2016. URL: https://arxiv.org/abs/1603.01512.
  28. Masahiro Ikeda, Atsushi Miyauchi, Yuuki Takai, and Yuichi Yoshida. Finding cheeger cuts in hypergraphs via heat equation. Theoretical Computer Science, 930:1-23, 2022. URL: https://doi.org/10.1016/J.TCS.2022.07.006.
  29. Mark Jerrum. Mathematical foundations of the markov chain monte carlo method. In Probabilistic methods for algorithmic discrete mathematics, pages 116-165. Springer, 1998. Google Scholar
  30. Mark Jerrum. Counting, sampling and integrating: algorithms and complexity. Springer Science & Business Media, 2003. Google Scholar
  31. Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and boolean isoperimetric-type theorems. SIAM Journal on Computing, 47(6):2238-2276, 2018. URL: https://doi.org/10.1137/16M1065872.
  32. Tsz Chiu Kwok, Lap Chi Lau, Yin Tat Lee, and Akshay Ramachandran. The paulsen problem, continuous operator scaling, and smoothed analysis. In Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 182-189, 2018. URL: https://doi.org/10.1145/3188745.3188794.
  33. Lap Chi Lau, Kam Chuen Tung, and Robert Wang. Cheeger inequalities for directed graphs and hypergraphs using reweighted eigenvalues. In n Proceedings of the ACM SIGACT Symposium on Theory of Computing (STOC), pages 1834-1847, 2023. URL: https://doi.org/10.1145/3564246.3585139.
  34. Lap Chi Lau, Kam Chuen Tung, and Robert Wang. Fast algorithms for directed graph partitioning using flows and reweighted eigenvalues. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 591-624, 2024. URL: https://doi.org/10.1137/1.9781611977912.22.
  35. David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. Google Scholar
  36. Pierre Mathieu. Log-sobolev and spectral gap inequalities for the knapsack markov chain. Markov Process. Relat. Fields, 8:595-610, 2002. Google Scholar
  37. Ravi Montenegro and Prasad Tetali. Mathematical aspects of mixing times in markov chains. Foundations and Trends® in Theoretical Computer Science, 1(3):237-354, 2006. URL: https://doi.org/10.1561/0400000003.
  38. Ben Morris and Alistair Sinclair. Random walks on truncated cubes and sampling 0-1 knapsack solutions. SIAM journal on computing, 34(1):195-226, 2004. URL: https://doi.org/10.1137/S0097539702411915.
  39. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. Google Scholar
  40. Ramesh Krishnan S Pallavoor, Sofya Raskhodnikova, and Erik Waingarten. Approximating the distance to monotonicity of boolean functions. Random Structures & Algorithms, 60(2):233-260, 2022. URL: https://doi.org/10.1002/RSA.21029.
  41. Romeo Rizzi and Alexandru I Tomescu. Faster fptases for counting and random generation of knapsack solutions. Information and Computation, 267:135-144, 2019. URL: https://doi.org/10.1016/J.IC.2019.04.001.
  42. Daniel Štefankovič, Santosh Vempala, and Eric Vigoda. A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM Journal on Computing, 41(2):356-366, 2012. URL: https://doi.org/10.1137/11083976X.
  43. Michel Talagrand. Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem. Geometric & Functional Analysis, 3(3):295-314, 1993. Google Scholar
  44. Gerald Teschl. Ordinary differential equations and dynamical systems, volume 140. American Mathematical Soc., 2012. Google Scholar
  45. Kam Chuen Tung. Reweighted Eigenvalues: A New Approach to Spectral Theory beyond Undirected Graphs. PhD thesis, University of Waterloo, 2025. Google Scholar
  46. Ramon van Handel. Probability in high dimension (lecture notes), 2014. Google Scholar
  47. Yuichi Yoshida. Nonlinear laplacian for digraphs and its applications to network analysis. In Proceedings of the Ninth ACM International Conference on Web Search and Data Mining, pages 483-492, 2016. URL: https://doi.org/10.1145/2835776.2835785.
  48. Yuichi Yoshida. Cheeger inequalities for submodular transformations. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2582-2601, 2019. URL: https://doi.org/10.1137/1.9781611975482.160.
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