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Tight Bounds for Sampling q-Colorings via Coupling from the Past

Authors Tianxing Ding , Hongyang Liu , Yitong Yin , Can Zhou

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LIPIcs.ICALP.2026.76.pdf
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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Markov-chain Monte Carlo methods
Keywords
  • perfect sampling
  • coupling from the past
  • graph coloring
  • bounding chains
  • Markov chains

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Abstract

The Coupling from the Past (CFTP) paradigm is a canonical method for perfect sampling. For uniform sampling of proper q-colorings in graphs with maximum degree Δ, the bounding chains of [Huber, STOC '98] provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime q ≥ (1+o(1))Δ². This was subsequently improved to q > 3Δ by [Bhandari and Chakraborty, STOC '20] and to q ≥ (8/3 + o(1))Δ by [Jain, Sah, and Sawhney, STOC '21].
In this work, we establish the asymptotically tight threshold for bounding-chain-based CFTP algorithms for graph colorings. We prove a lower bound showing that all such algorithms satisfying the standard contraction property require q ≥ 2.5Δ, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold q ≥ (2.5 + o(1))Δ via an optimal design of bounding chains.

Cite As Get BibTex

Tianxing Ding, Hongyang Liu, Yitong Yin, and Can Zhou. Tight Bounds for Sampling q-Colorings via Coupling from the Past. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 76:1-76:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.76

Author Details

Tianxing Ding
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China
Hongyang Liu
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China
Yitong Yin
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China
Can Zhou
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, China

References

  1. Konrad Anand, Andreas Göbel, Marcus Pappik, and Will Perkins. Perfect sampling for hard spheres from strong spatial mixing. In Nicole Megow and Adam D. Smith, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, Atlanta, Georgia, USA, September 11-13, 2023, LIPIcs, pages 38:1-38:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.38.
  2. Konrad Anand and Mark Jerrum. Perfect sampling in infinite spin systems via strong spatial mixing. SIAM Journal on Computing, 51(4):1280-1295, 2022. URL: https://doi.org/10.1137/21M1437433.
  3. Ferenc Bencs, Khallil Berrekkal, and Guus Regts. Deterministic approximate counting of colorings with fewer than 2δ colors via absence of zeros. TheoretiCS, 5:1:1-1:41, 2026. URL: https://doi.org/10.46298/theoretics.26.1.
  4. Siddharth Bhandari and Sayantan Chakraborty. Improved bounds for perfect sampling of k-colorings in graphs. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 631-642, 2020. URL: https://doi.org/10.1137/20M1366666.
  5. Russ Bubley and Martin E. Dyer. Path coupling: A technique for proving rapid mixing in markov chains. In 38th Annual Symposium on Foundations of Computer Science, FOCS 1997, Miami Beach, Florida, USA, October 19-22, 1997, pages 223-231. IEEE Computer Society, 1997. URL: https://doi.org/10.1109/SFCS.1997.646111.
  6. Charlie Carlson and Eric Vigoda. Flip dynamics for sampling colorings: Improving (11/6—ε) using a simple metric. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2194-2212. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.71.
  7. Sitan Chen, Michelle Delcourt, Ankur Moitra, Guillem Perarnau, and Luke Postle. Improved bounds for randomly sampling colorings via linear programming. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2216-2234, 2019. URL: https://doi.org/10.1137/1.9781611975482.134.
  8. Xiaoyu Chen, Weiming Feng, Heng Guo, Xinyuan Zhang, and Zongrui Zou. Deterministic counting from coupling independence. In 66th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2025, Sydney, Australia, December 14-17, 2025, pages 667-683. IEEE, 2025. URL: https://doi.org/10.1109/FOCS63196.2025.00036.
  9. Xiaoyu Chen, Weiming Feng, Yitong Yin, and Xinyuan Zhang. Rapid mixing of glauber dynamics via spectral independence for all degrees. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 137-148. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00022.
  10. Zongchen Chen, Kuikui Liu, Nitya Mani, and Ankur Moitra. Strong spatial mixing for colorings on trees and its algorithmic applications. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 810-845, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00053.
  11. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion. In STOC, pages 1537-1550. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451035.
  12. Henry Cohn, Robin Pemantle, and James Gary Propp. Generating a random sink-free orientation in quadratic time. Electron. J. Comb., 9(1), 2002. URL: https://doi.org/10.37236/1627.
  13. Martin Dyer and Catherine Greenhill. Random walks on combinatorial objects. London Mathematical Society Lecture Note Series, pages 101-136, 1999. Google Scholar
  14. Martin E. Dyer, Alan M. Frieze, Thomas P. Hayes, and Eric Vigoda. Randomly coloring constant degree graphs. Random Struct. Algorithms, 43(2):181-200, 2013. URL: https://doi.org/10.1002/RSA.20451.
  15. Weiming Feng, Heng Guo, Chunyang Wang, Jiaheng Wang, and Yitong Yin. Towards derandomising markov chain monte carlo. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1963-1990. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00120.
  16. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Rapid mixing from spectral independence beyond the boolean domain. ACM Trans. Algorithms, 18(3):28:1-28:32, 2022. URL: https://doi.org/10.1145/3531008.
  17. Weiming Feng, Nisheeth K. Vishnoi, and Yitong Yin. Dynamic sampling from graphical models. SIAM J. Comput., 50(2):350-381, 2021. URL: https://doi.org/10.1137/20M1315099.
  18. James Allen Fill and Mark Huber. The randomness recycler: A new technique for perfect sampling. In 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, Redondo Beach, California, USA, November 12-14, 2000, pages 503-511. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/SFCS.2000.892138.
  19. James Allen Fill, Motoya Machida, Duncan J. Murdoch, and Jeffrey S. Rosenthal. Extension of fill’s perfect rejection sampling algorithm to general chains. Random Struct. Algorithms, 17(3-4):290-316, 2000. URL: https://doi.org/10.1002/1098-2418(200010/12)17:3/4%3C290::AID-RSA6%3E3.0.CO;2-Q.
  20. Andreas Galanis, Daniel Stefankovic, and Eric Vigoda. Inapproximability for antiferromagnetic spin systems in the tree nonuniqueness region. J. ACM, 62(6):50:1-50:60, 2015. URL: https://doi.org/10.1145/2785964.
  21. David Gamarnik and Dmitriy Katz. Correlation decay and deterministic FPTAS for counting colorings of a graph. J. Discrete Algorithms, 12:29-47, 2012. URL: https://doi.org/10.1016/J.JDA.2010.10.002.
  22. Heng Guo, Mark Jerrum, and Jingcheng Liu. Uniform sampling through the lovász local lemma. J. ACM, 66(3):18:1-18:31, 2019. URL: https://doi.org/10.1145/3310131.
  23. Andreas Göbel, Jingcheng Liu, Pasin Manurangsi, and Marcus Pappik. Perfect sampling from rapidly mixing markov chains, 2024. URL: https://arxiv.org/abs/2410.00882.
  24. Olle Häggström and Karin Nelander. On exact simulation of markov random fields using coupling from the past. Scandinavian Journal of Statistics, 26(3):395-411, 1999. URL: https://doi.org/10.1111/1467-9469.00156.
  25. Thomas P. Hayes. A simple condition implying rapid mixing of single-site dynamics on spin systems. In 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, Berkeley, California, USA, October 21-24, 2006, Proceedings, pages 39-46. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/FOCS.2006.6.
  26. Thomas P. Hayes and Eric Vigoda. A non-markovian coupling for randomly sampling colorings. In 44th Symposium on Foundations of Computer Science, FOCS 2003, Cambridge, MA, USA, October 11-14, 2003, Proceedings, pages 618-627. IEEE Computer Society, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238234.
  27. Thomas P. Hayes and Eric Vigoda. Coupling with the stationary distribution and improved sampling for colorings and independent sets. Ann. Appl. Probab., 16(3):1297-1318, 2006. Google Scholar
  28. Kun He, Xiaoming Sun, and Kewen Wu. Perfect sampling for (atomic) lovász local lemma, 2021. URL: https://arxiv.org/abs/2107.03932.
  29. Kun He, Chunyang Wang, and Yitong Yin. Sampling lovász local lemma for general constraint satisfaction solutions in near-linear time. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 147-158. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00021.
  30. Mark Huber. Exact sampling and approximate counting techniques. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 31-40. ACM, 1998. URL: https://doi.org/10.1145/276698.276709.
  31. Mark L. Huber. Perfect sampling using bounding chains. The Annals of Applied Probability, 14(2):735-753, 2004. URL: https://doi.org/10.1214/105051604000000080.
  32. M.L. Huber. Perfect Simulation. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. CRC Press, 2016. URL: https://books.google.com.tw/books?id=xD5qCwAAQBAJ.
  33. O. Häggström and K. Nelander. Exact sampling from anti-monotone systems. Statistica Neerlandica, 52(3):360-380, 1998. URL: https://doi.org/10.1111/1467-9574.00090.
  34. Vishesh Jain, Huy Tuan Pham, and Thuy-Duong Vuong. Spectral independence, coupling, and the spectral gap of the glauber dynamics. Inf. Process. Lett., 177:106268, 2022. URL: https://doi.org/10.1016/J.IPL.2022.106268.
  35. Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney. Perfectly sampling k ≥ (8/3 + o(1))Δ-colorings in graphs. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1589-1600. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451012.
  36. Mark Jerrum. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Struct. Algorithms, 7(2):157-166, 1995. URL: https://doi.org/10.1002/RSA.3240070205.
  37. Mark Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci., 43:169-188, 1986. URL: https://doi.org/10.1016/0304-3975(86)90174-X.
  38. Johan Jonasson. Uniqueness of uniform random colorings of regular trees. Statistics & Probability Letters, 57(3):243-248, 2002. URL: https://doi.org/10.1016/S0167-7152(02)00054-8.
  39. Hongyang Liu, Chunyang Wang, and Yitong Yin. Local gibbs sampling beyond local uniformity. In Kasper Green Larsen and Barna Saha, editors, Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2026, Vancouver, BC, Canada, January 11-14, 2026, pages 996-1025. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.41.
  40. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. A deterministic algorithm for counting colorings with 2-delta colors. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 1380-1404. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00085.
  41. Jingcheng Liu, Chunyang Wang, Yitong Yin, and Yixiao Yu. Phase transitions via complex extensions of markov chains. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 903-914. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718230.
  42. Pinyan Lu and Yitong Yin. Improved FPTAS for multi-spin systems. In Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, Lecture Notes in Computer Science, pages 639-654. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_44.
  43. James Gary Propp and David Bruce Wilson. Exact sampling with coupled markov chains and applications to statistical mechanics. Random Struct. Algorithms, 9(1-2):223-252, 1996. URL: https://doi.org/10.1002/(SICI)1098-2418(199608/09)9:1/2%3C223::AID-RSA14%3E3.0.CO;2-O.
  44. Guoliang Qiu, Yanheng Wang, and Chihao Zhang. A perfect sampler for hypergraph independent sets. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, Paris, France, July 4-8, 2022, LIPIcs, pages 103:1-103:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.103.
  45. Jesús Salas and Alan D Sokal. Absence of phase transition for antiferromagnetic potts models via the dobrushin uniqueness theorem. Journal of Statistical Physics, 86:551-579, 1997. Google Scholar
  46. Eric Vigoda. Improved bounds for sampling colorings. In 40th Annual Symposium on Foundations of Computer Science, FOCS 1999, New York, NY, USA, October 17-18, 1999, pages 51-59. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814577.
  47. David Bruce Wilson. Generating random spanning trees more quickly than the cover time. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 296-303. ACM, 1996. URL: https://doi.org/10.1145/237814.237880.
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