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Near-Linear Time Samplers for Matroid Independent Sets with Applications

Authors Xiaoyu Chen, Heng Guo, Xinyuan Zhang, Zongrui Zou

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Author Details

Xiaoyu Chen
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, 163 Xianlin Avenue, Nanjing, Jiangsu Province, China
Heng Guo
  • School of Informatics, University of Edinburgh, Informatics Forum, 10 Crichton Street, Edinburgh, EH8 9AB, UK
Xinyuan Zhang
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, 163 Xianlin Avenue, Nanjing, Jiangsu Province, China
Zongrui Zou
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, 163 Xianlin Avenue, Nanjing, Jiangsu Province, China

Funding

  • Guo, Heng: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947778).

Acknowledgements

We would like to thank the hospitality of NII Shonan meeting No. 186, where some of the discussion took place.

Cite As Get BibTex

Xiaoyu Chen, Heng Guo, Xinyuan Zhang, and Zongrui Zou. Near-Linear Time Samplers for Matroid Independent Sets with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.32

Abstract

We give a Õ(n) time almost uniform sampler for independent sets of a matroid, whose ground set has n elements and is given by an independence oracle. As a consequence, one can sample connected spanning subgraphs of a given graph G = (V,E) in Õ(|E|) time, whereas the previous best algorithm takes O(|E||V|) time. This improvement, in turn, leads to a faster running time on estimating all-terminal network reliability. Furthermore, we generalise this near-linear time sampler to the random cluster model with q ≤ 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Network reliability
  • Random cluster modek
  • Matroid
  • Bases-exchange walk

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References

  1. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials III: Mason’s ultra-log-concavity conjecture for independent sets of matroids. arXiv, abs/1811.01600, 2018. URL: https://arxiv.org/abs/1811.01600.
  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 STOC, pages 1-12. ACM, 2019. Google Scholar
  3. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant, and Thuy-Duong Vuong. Log-concave polynomials IV: approximate exchange, tight mixing times, and near-optimal sampling of forests. In STOC, pages 408-420. ACM, 2021. URL: https://arxiv.org/abs/2004.07220v2.
  4. Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids. In FOCS, pages 35-46. IEEE, 2018. URL: https://arxiv.org/abs/1807.00929v2.
  5. Petter Brändén and June Huh. Hodge-Riemann relations for Potts model partition functions. arXiv, abs/1811.01696, 2018. URL: https://arxiv.org/abs/1811.01696.
  6. Petter Brändén and June Huh. Lorentzian polynomials. Ann. of Math. (2), 192(3):821-891, 2020. URL: https://arxiv.org/abs/1902.03719v6.
  7. Mary Cryan, Heng Guo, and Giorgos Mousa. Modified log-Sobolev inequalities for strongly log-concave distributions. Ann. Probab., 49(1):506-525, 2021. Google Scholar
  8. Tomás Feder and Milena Mihail. Balanced matroids. In STOC, pages 26-38. ACM, 1992. Google Scholar
  9. Cornelius M. Fortuin and Pieter W. Kasteleyn. On the random-cluster model. I. Introduction and relation to other models. Physica, 57:536-564, 1972. Google Scholar
  10. Heng Guo and Kun He. Tight bounds for popping algorithms. Random Structures Algorithms, 57(2):371-392, 2020. Google Scholar
  11. Heng Guo and Mark Jerrum. A polynomial-time approximation algorithm for all-terminal network reliability. SIAM J. Comput., 48(3):964-978, 2019. Google Scholar
  12. Heng Guo, Mark Jerrum, and Jingcheng Liu. Uniform sampling through the Lovász local lemma. J. ACM, 66(3):18:1-18:31, 2019. Google Scholar
  13. Leonid Gurvits. On multivariate Newton-like inequalities. In Advances in combinatorial mathematics, pages 61-78. Springer, Berlin, 2009. Google Scholar
  14. Leonid Gurvits. A polynomial-time algorithm to approximate the mixed volume within a simply exponential factor. Discrete Comput. Geom., 41(4):533-555, 2009. Google Scholar
  15. Mark Jerrum. On the complexity of evaluating multivariate polynomials. PhD thesis, The University of Edinburgh, 1981. Google Scholar
  16. Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci., 43(2-3):169-188, 1986. Google Scholar
  17. Vladimir Kolmogorov. A faster approximation algorithm for the gibbs partition function. In COLT, volume 75, pages 228-249. PMLR, 2018. Google Scholar
  18. Giorgos Mousa. Local-to-Global Functional Inequalities in Simplicial Complexes. PhD thesis, The University of Edinburgh, 2022. Google Scholar
  19. J. Scott Provan and Michael O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput., 12(4):777-788, 1983. Google Scholar
  20. Daniel Štefankovič, Santosh Vempala, and Eric Vigoda. Adaptive simulated annealing: a near-optimal connection between sampling and counting. J. ACM, 56(3):Art. 18, 36, 2009. Google Scholar
  21. Leslie G. Valiant. The complexity of enumeration and reliability problems. SIAM J. Comput., 8(3):410-421, 1979. Google Scholar
  22. Christian Wulff-Nilsen. Faster deterministic fully-dynamic graph connectivity. In SODA, pages 1757-1769. SIAM, 2013. Google Scholar
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