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Improved Bounds for Randomly Colouring Simple Hypergraphs

Authors Weiming Feng , Heng Guo , Jiaheng Wang

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Author Details

Weiming Feng
  • School of Informatics, University of Edinburgh, UK
Heng Guo
  • School of Informatics, University of Edinburgh, UK
Jiaheng Wang
  • School of Informatics, University of Edinburgh, UK

Funding

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).

Cite AsGet BibTex

Weiming Feng, Heng Guo, and Jiaheng Wang. Improved Bounds for Randomly Colouring Simple Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.25

Abstract

We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree Δ. For any δ > 0, if k ≥ 20(1+δ)/δ and q ≥ 100Δ^({2+δ}/{k-4/δ-4}), the running time of our algorithm is Õ(poly(Δ k)⋅ n^1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021), and does not require Ω(log n) colours unlike the work of Frieze and Anastos (2017).

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Approximate counting
  • Markov chain
  • Mixing time
  • Hypergraph colouring

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References

  1. Noga Alon. A parallel algorithmic version of the local lemma. Random Struct. Algorithms, 2(4):367-378, 1991. Google Scholar
  2. József Beck. An algorithmic approach to the Lovász local lemma. I. Random Struct. Algorithms, 2(4):343-366, 1991. Google Scholar
  3. Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Daniel Štefankovič. Approximation via correlation decay when strong spatial mixing fails. SIAM J. Comput., 48(2):279-349, 2019. Google Scholar
  4. Christian Borgs, Jennifer Chayes, Jeff Kahn, and László Lovász. Left and right convergence of graphs with bounded degree. Random Struct. Algorithms, 42(1):1-28, 2013. Google Scholar
  5. Artur Czumaj and Christian Scheideler. Coloring nonuniform hypergraphs: A new algorithmic approach to the general Lovász local lemma. Random Struct. Algorithms, 17(3-4):213-237, 2000. Google Scholar
  6. P. Erdős and L. Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, pages 609-627. Colloq. Math. Soc. János Bolyai, Vol. 10. North-Holland, 1975. Google Scholar
  7. Weiming Feng, Heng Guo, and Jiaheng Wang. Improved bounds for randomly colouring simple hypergraphs. CoRR, abs/2202.05554, 2022. URL: http://arxiv.org/abs/2202.05554.
  8. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Fast sampling and counting k-SAT solutions in the local lemma regime. J. ACM, 68(6):40:1-40:42, 2021. Google Scholar
  9. Weiming Feng, Kun He, and Yitong Yin. Sampling constraint satisfaction solutions in the local lemma regime. arXiv, abs/2011.03915, 2020. URL: http://arxiv.org/abs/2011.03915.
  10. Alan Frieze and Dhruv Mubayi. Coloring simple hypergraphs. J. Combin. Theory Ser. B, 103(6):767-794, 2013. Google Scholar
  11. Alan M. Frieze and Michael Anastos. Randomly coloring simple hypergraphs with fewer colors. Inf. Process. Lett., 126:39-42, 2017. URL: https://doi.org/10.1016/j.ipl.2017.06.005.
  12. Alan M. Frieze and Páll Melsted. Randomly coloring simple hypergraphs. Inf. Process. Lett., 111(17):848-853, 2011. URL: https://doi.org/10.1016/j.ipl.2011.06.001.
  13. Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Kuan Yang. Counting solutions to random CNF formulas. SIAM J. Comput., 50(6):1701-1738, 2021. Google Scholar
  14. Andreas Galanis, Heng Guo, and Jiaheng Wang. Inapproximability of counting hypergraph colourings. arXiv preprint, 2021. URL: http://arxiv.org/abs/2107.05486.
  15. Heidi Gebauer, Tibor Szabó, and Gábor Tardos. The local lemma is asymptotically tight for SAT. J. ACM, 63(5):43:1-43:32, 2016. Google Scholar
  16. 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
  17. Heng Guo, Chao Liao, Pinyan Lu, and Chihao Zhang. Counting hypergraph colorings in the local lemma regime. SIAM J. Comput., 48(4):1397-1424, 2019. Google Scholar
  18. Bernhard Haeupler, Barna Saha, and Aravind Srinivasan. New constructive aspects of the Lovász local lemma. J. ACM, 58(6):28, 2011. Google Scholar
  19. Kun He, Xiaoming Sun, and Kewen Wu. Perfect sampling for (atomic) Lovász local lemma. arXiv, abs/2107.03932, 2021. URL: http://arxiv.org/abs/2107.03932.
  20. Jonathan Hermon, Allan Sly, and Yumeng Zhang. Rapid mixing of hypergraph independent sets. Random Struct. Algorithms, 54(4):730-767, 2019. Google Scholar
  21. Mark Huber. Exact sampling and approximate counting techniques. In STOC, pages 31-40. ACM, 1998. Google Scholar
  22. Mark Huber. Approximation algorithms for the normalizing constant of Gibbs distributions. Ann. Appl. Probab., 25(2):974-985, 2015. Google Scholar
  23. Vishesh Jain, Huy Tuan Pham, and Thuy Duong Vuong. Towards the sampling Lovász local lemma. arXiv, abs/2011.12196, 2020. URL: http://arxiv.org/abs/2011.12196.
  24. Vishesh Jain, Huy Tuan Pham, and Thuy Duong Vuong. On the sampling Lovász local lemma for atomic constraint satisfaction problems. arXiv, abs/2102.08342, 2021. URL: http://arxiv.org/abs/2102.08342.
  25. Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci., 43:169-188, 1986. Google Scholar
  26. Vladimir Kolmogorov. A faster approximation algorithm for the Gibbs partition function. In COLT, pages 228-249. PMLR, 2018. URL: http://proceedings.mlr.press/v75/kolmogorov18a.html.
  27. Ankur Moitra. Approximate counting, the Lovász local lemma, and inference in graphical models. J. ACM, 66(2):10:1-10:25, 2019. Google Scholar
  28. Michael Molloy and Bruce A. Reed. Further algorithmic aspects of the local lemma. In STOC, pages 524-529. ACM, 1998. Google Scholar
  29. Robin A. Moser. A constructive proof of the Lovász local lemma. In STOC, pages 343-350. ACM, 2009. Google Scholar
  30. Robin A. Moser and Gábor Tardos. A constructive proof of the general Lovász local lemma. J. ACM, 57(2):11, 2010. Google Scholar
  31. James G. Propp and David B. Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms, 9(1-2):223-252, 1996. Google Scholar
  32. Aravind Srinivasan. Improved algorithmic versions of the Lovász local lemma. In SODA, pages 611-620. SIAM, 2008. Google Scholar
  33. Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. Google Scholar
  34. Daniel Štefankovič, Santosh Vempala, and Eric Vigoda. Adaptive simulated annealing: A near-optimal connection between sampling and counting. J. ACM, 56(3):18, 2009. Google Scholar
  35. Ian M Wanless and David R Wood. A general framework for hypergraph colouring. arXiv preprint, 2020. URL: http://arxiv.org/abs/2008.00775.
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