Sampling List Packings

Authors Evan Camrud , Ewan Davies , Alex Karduna, Holden Lee



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2025.24.pdf
  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

Author Details

Evan Camrud
  • Department of Mathematics and Statistics, Middlebury College, VT, USA
Ewan Davies
  • Department of Computer Science, Colorado State University, Fort Collins, CO, USA
Alex Karduna
  • Department of Computer Science, Colorado State University, Fort Collins, CO, USA
Holden Lee
  • Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA

Acknowledgements

We thank Colin McSwiggen and Semon Rezchikov for organizing Random Theory 2023, where this collaboration began. We thank Charlie Carlson and Guillem Perarnau for comments that led to improvements to the exposition of the paper.

Cite As Get BibTex

Evan Camrud, Ewan Davies, Alex Karduna, and Holden Lee. Sampling List Packings. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.24

Abstract

We initiate the study of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted substantial attention. For list packing the setup is similar, but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. The existence of a list packing implies the existence of a list coloring, but the converse is false. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting and sampling. 
In graphs of maximum degree Δ and when the number of colors is at least Ω(Δ²), we give a fully polynomial-time randomized approximation scheme (FPRAS) based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Approximation algorithms
Keywords
  • List packing
  • Graph colouring
  • Markov chains
  • Path coupling

Metrics

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

References

  1. Noga Alon, Stijn Cambie, and Ross J. Kang. Asymmetric List Sizes in Bipartite Graphs. Annals of Combinatorics, 25(4):913-933, 2021. URL: https://doi.org/10.1007/s00026-021-00552-5.
  2. Ferenc Bencs, Ewan Davies, Viresh Patel, and Guus Regts. On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs. Annales de l'Institut Henri Poincaré D, 8(3):459-489, 2021. URL: https://doi.org/10.4171/AIHPD/108.
  3. 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, STOC 2020, pages 631-642, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384244.
  4. Peter Bradshaw. Graph Colorings with Local Restrictions. PhD thesis, Simon Fraser University, 2022. URL: https://summit.sfu.ca/item/35851.
  5. R. Bubley and M. Dyer. Path coupling: A technique for proving rapid mixing in Markov chains. In Proceedings 38th Annual Symposium on Foundations of Computer Science, pages 223-231, Miami Beach, FL, USA, 1997. IEEE Comput. Soc. URL: https://doi.org/10.1109/SFCS.1997.646111.
  6. Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, and Ross J. Kang. Packing list-colorings. Random Structures & Algorithms, 64(1):62-93, 2024. URL: https://doi.org/10.1002/rsa.21181.
  7. Stijn Cambie, Wouter Cames van Batenburg, and Xuding Zhu. Disjoint list-colorings for planar graphs. arXiv preprint, 2023. URL: https://arxiv.org/abs/2312.17233.
  8. Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, and Ross J. Kang. List packing number of bounded degree graphs. Combinatorics, Probability and Computing, 33(6):807-828, 2024. URL: https://doi.org/10.1017/S0963548324000191.
  9. Charlie Carlson and Eric Vigoda. Flip Dynamics for Sampling Colorings: Improving (11/6-ε) Using a Simple Metric. To appear in SODA 2025, 2025. URL: https://doi.org/10.48550/arXiv.2407.04870.
  10. 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, Proceedings, pages 2216-2234. Society for Industrial and Applied Mathematics, 2019. URL: https://doi.org/10.1137/1.9781611975482.134.
  11. Martin Dyer and Catherine Greenhill. A more rapidly mixing Markov chain for graph colorings. Random Structures & Algorithms, 13(3-4):285-317, 1998. URL: https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4<285::AID-RSA6>3.0.CO;2-R.
  12. Martin Dyer and Catherine Greenhill. Random Walks on Combinatorial Objects. In J. D. Lamb and D. A. Preece, editors, Surveys in Combinatorics, 1999, pages 101-136. Cambridge University Press, 1 edition, 1999. URL: https://doi.org/10.1017/CBO9780511721335.005.
  13. Paul Erdős, Arthur L. Rubin, and Herbert Taylor. Choosability in graphs. In Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979), Congress. Numer., XXVI, pages 125-157. Utilitas Math., Winnipeg, Man., 1980. Google Scholar
  14. P. Hall. On Representatives of Subsets. Journal of the London Mathematical Society, s1-10(1):26-30, 1935. URL: https://doi.org/10.1112/jlms/s1-10.37.26.
  15. Mark Huber. Exact sampling and approximate counting techniques. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 31-40, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276709.
  16. Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney. Perfectly Sampling k ≥ (8/3 + o(1))Δ-colorings in Graphs. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1589-1600, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451012.
  17. Mark Jerrum. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures & Algorithms, 7(2):157-165, 1995. URL: https://doi.org/10.1002/rsa.3240070205.
  18. Mark Jerrum. Counting, Sampling and Integrating: Algorithms and Complexity. Lectures in Mathematics ETH Zürich. Birkhauser Verlag, Basel Boston, MA, 2003. Google Scholar
  19. Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, 43:169-188, 1986. URL: https://doi.org/10.1016/0304-3975(86)90174-X.
  20. Hemanshu Kaul, Rogers Mathew, Jeffrey A. Mudrock, and Michael J. Pelsmajer. Flexible list colorings: Maximizing the number of requests satisfied. Journal of Graph Theory, 106(4):887-906, 2024. URL: https://doi.org/10.1002/jgt.23103.
  21. Hemanshu Kaul and Jeffrey A. Mudrock. Counting Packings of List-colorings of Graphs. arXiv preprint, 2024. URL: https://arxiv.org/abs/2401.11025.
  22. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. A Deterministic Algorithm for Counting Colorings with 2-Delta Colors. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1380-1404, Baltimore, MD, USA, 2019. IEEE. URL: https://doi.org/10.1109/FOCS.2019.00085.
  23. Pinyan Lu and Yitong Yin. Improved FPTAS for Multi-spin Systems. In David Hutchison, Takeo Kanade, Josef Kittler, Jon M. Kleinberg, Friedemann Mattern, John C. Mitchell, Moni Naor, Oscar Nierstrasz, C. Pandu Rangan, Bernhard Steffen, Madhu Sudan, Demetri Terzopoulos, Doug Tygar, Moshe Y. Vardi, Gerhard Weikum, Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 8096, pages 639-654. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_44.
  24. Kyle MacKeigan. Independent coverings and orthogonal colourings. Discrete Mathematics, 344(8):112431, 2021. URL: https://doi.org/10.1016/j.disc.2021.112431.
  25. Jesús Salas and Alan D. Sokal. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. Journal of Statistical Physics, 86(3):551-579, 1997. URL: https://doi.org/10.1007/BF02199113.
  26. E. Vigoda. Improved bounds for sampling colorings. In 40th Annual Symposium on Foundations of Computer Science, pages 51-59, New York City, NY, USA, 1999. IEEE Comput. Soc. URL: https://doi.org/10.1109/SFFCS.1999.814577.
  27. Vadim G Vizing. Coloring the vertices of a graph in prescribed colors. Diskret. Analiz, 29(3):10, 1976. 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