Document Open Access Logo

Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size

Authors Vishesh Jain , Clayton Mizgerd

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2024.63.pdf
  • Filesize: 0.67 MB
  • 13 pages

Document Identifiers

Author Details

Vishesh Jain
  • Department of Mathematics, Statistics, and Computer Science, University of Illinois Chicago, Chicago, IL, USA
Clayton Mizgerd
  • Department of Mathematics, Statistics, and Computer Science, University of Illinois Chicago, Chicago, IL, USA

Funding

  • Jain, Vishesh: NSF CAREER award DMS-2237646
  • Mizgerd, Clayton: NSF award ECCS-2217023

Acknowledgements

We thank Huy Tuan Pham for helpful discussions and anonymous referees for several useful suggestions.

Cite As Get BibTex

Vishesh Jain and Clayton Mizgerd. Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.63

Abstract

Let G = (V,E) be a graph on n vertices and let m^*(G) denote the size of a maximum matching in G. We show that for any δ > 0 and for any 1 ≤ k ≤ (1-δ)m^*(G), the down-up walk on matchings of size k in G mixes in time polynomial in n. Previously, polynomial mixing was not known even for graphs with maximum degree Δ, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time O_{Δ,δ}(nlog{n}). 
In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov-chain Monte Carlo methods
Keywords
  • Down-up walk
  • Matchings
  • MCMC

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Yeganeh Alimohammadi, Nima Anari, Kirankumar Shiragur, and Thuy-Duong Vuong. Fractionally log-concave and sector-stable polynomials: counting planar matchings and more. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 433-446, 2021. Google Scholar
  2. Nima Anari, Vishesh Jain, Frederic Koehler, Huy Tuan Pham, and Thuy-Duong Vuong. Universality of spectral independence with applications to fast mixing in spin glasses. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5029-5056. SIAM, 2024. Google Scholar
  3. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In IEEE Symposium on Foundations of Computer Science, 2020. Google Scholar
  4. Nima Anari, Yang P Liu, and Thuy-Duong Vuong. Optimal sublinear sampling of spanning trees and determinantal point processes via average-case entropic independence. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 123-134. IEEE, 2022. Google Scholar
  5. Pietro Caputo. Spectral gap inequalities in product spaces with conservation laws. In Stochastic analysis on large scale interacting systems, volume 39, pages 53-89. Mathematical Society of Japan, 2004. Google Scholar
  6. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of glauber dynamics: Entropy factorization via high-dimensional expansion. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1537-1550, 2021. Google Scholar
  7. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Spectral independence via stability and applications to holant-type problems. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 149-160. IEEE, 2022. Google Scholar
  8. Paul Dagum, Michael Luby, Milena Mihail, and U Vazirani. Polytopes, permanents and graphs with large factors. In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pages 412-421. IEEE Computer Society, 1988. Google Scholar
  9. Ewan Davies and Will Perkins. Approximately counting independent sets of a given size in bounded-degree graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP), volume 198, pages 62:1-62:18, 2021. Google Scholar
  10. Persi Diaconis and Mehrdad Shahshahani. Time to reach stationarity in the bernoulli-laplace diffusion model. SIAM Journal on Mathematical Analysis, 18(1):208-218, 1987. Google Scholar
  11. Persi Diaconis and Daniel Stroock. Geometric bounds for eigenvalues of markov chains. The annals of applied probability, pages 36-61, 1991. Google Scholar
  12. Yuval Filmus, Ryan O’Donnell, and Xinyu Wu. Log-sobolev inequality for the multislice, with applications. Electronic Journal of Probability, 27:1-30, 2022. Google Scholar
  13. Vishesh Jain, Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong. Optimal mixing of the down-up walk on independent sets of a given size. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1665-1681. IEEE, 2023. Google Scholar
  14. Vishesh Jain, Will Perkins, Ashwin Sah, and Mehtaab Sawhney. Approximate counting and sampling via local central limit theorems. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1473-1486, 2022. Google Scholar
  15. Mark Jerrum. Two-dimensional monomer-dimer systems are computationally intractable. Journal of Statistical Physics, 48:121-134, 1987. Google Scholar
  16. Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-1178, 1989. Google Scholar
  17. Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM (JACM), 51(4):671-697, 2004. Google Scholar
  18. Pieter Kasteleyn. Graph theory and crystal physics. Graph Theory and Theoretical Physics, pages 43-110, 1967. Google Scholar
  19. Tzong-Yow Lee and Horng-Tzer Yau. Logarithmic sobolev inequality for some models of random walks. The Annals of Probability, 26(4):1855-1873, 1998. Google Scholar
  20. David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. Google Scholar
  21. Kuikui Liu. From coupling to spectral independence and blackbox comparison with the down-up walk. arXiv preprint, 2021. URL: https://arxiv.org/abs/2103.11609.
  22. Alistair Sinclair. Improved bounds for mixing rates of markov chains and multicommodity flow. Combinatorics, probability and Computing, 1(4):351-370, 1992. Google Scholar
  23. Leslie G Valiant. The complexity of computing the permanent. Theoretical computer science, 8(2):189-201, 1979. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact