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Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm

Author Sam Olesker-Taylor

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Sam Olesker-Taylor
  • Department of Statistics, University of Warwick, UK

Acknowledgements

The author would like to thank Frank Kelly for multiple detailed discussions on this topic, as well as reading through the paper. Additionally, thanks go to Perla Sousi and Luca Zanetti, who also read the paper. Their feedback has been invaluable in preparing this manuscript.

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Sam Olesker-Taylor. Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.20

Abstract

In the hardcore model, certain vertices in a graph are active: the active vertices must form an independent set. We extend this to a multicoloured version: instead of simply being active or not, the active vertices are assigned a colour; active vertices of the same colour must not be adjacent. This models a scenario in which two neighbouring resources may interfere when active - eg, short-range radio communication. However, there are multiple channels (colours) available; they only interfere if both use the same channel. Other applications include routing in fibreoptic networks. We analyse Glauber dynamics. Vertices update their status at random times, at which a uniform colour is proposed: the vertex is assigned that colour if it is available; otherwise, it is set inactive. We find conditions for fast mixing of these dynamics. We also use them to model a queueing system: vertices only serve customers in their queue whilst active. The mixing estimates are applied to establish positive recurrence of the queue lengths, and bound their expectation in equilibrium.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • mixing time
  • queueing theory
  • hardcore model
  • proper colourings
  • independent set
  • data transmission
  • randomised algorithms
  • routing
  • scheduling
  • multihop wireless networks

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References

  1. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pages 1319-1330. IEEE Computer Soc., Los Alamitos, CA, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00125.
  2. Nayantara Bhatnagar, Allan Sly, and Prasad Tetali. Decay of correlations for the hardcore model on the d-regular random graph. Electron. J. Probab., 21:Paper No. 9, 42 pp., 2016. URL: https://doi.org/10.1214/16-EJP3552.
  3. R. Boorstyn, A. Kershenbaum, B. Maglaris, and V. Sahin. Throughput analysis in multihop csma packet radio networks. IEEE Transactions on Communications, 35(3):267-274, March 1987. URL: https://doi.org/10.1109/TCOM.1987.1096769.
  4. Torsten Braun, Andreas Kassler, Maria Kihl, Veselin Rakocevic, Vasilios Siris, and Geert Heijenk. Multihop wireless networks. In Vasilios Siris, Torsten Braun, Francisco Barcelo-Arroyo, Dirk Staehle, Giovanni Giambene, and Yevgeni Koucheryavy, editors, Traffic and QoS Management in Wireless Multimedia Networks: COST 290 Final Report, pages 201-265. Springer US, Boston, MA, 2009. Google Scholar
  5. Raffaele Bruno and Maddalena Nurchis. Survey on diversity-based routing in wireless mesh networks: Challenges and solutions. Computer Communications, 33(3):269-282, February 2010. URL: https://doi.org/10.1016/j.comcom.2009.09.003.
  6. Ross Bubley and Martin Dyer. Path coupling: A technique for proving rapid mixing in markov chains. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science, FOCS '97, pages 223-, Washington, DC, USA, 1997. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.1997.646111.
  7. Andreas Galanis, Qi Ge, Daniel Štefankovič, Eric Vigoda, and Linji Yang. Improved inapproximability results for counting independent sets in the hard-core model. Random Structures Algorithms, 45(1):78-110, 2014. URL: https://doi.org/10.1002/rsa.20479.
  8. Song-Nam Hong and Ivana Marić. Multihop wireless backhaul for 5g. In Ivana Marić, Shlomo Shamai (Shitz), and Osvaldo Simeone, editors, Information Theoretic Perspectives on 5G Systems and Beyond, pages 131-165. Cambridge University Press, Cambridge, 2022. URL: https://doi.org/10.1017/9781108241267.004.
  9. L. Jiang and J. Walrand. A distributed csma algorithm for throughput and utility maximization in wireless networks. IEEE/ACM Transactions on Networking, 18(3):960-972, June 2010. URL: https://doi.org/10.1109/TNET.2009.2035046.
  10. Libin Jiang, Mathieu Leconte, Jian Ni, Rayadurgam Srikant, and Jean Walrand. Fast mixing of parallel glauber dynamics and low-delay csma scheduling. IEEE Transactions on Information Theory, 58(10):6541-6555, October 2012. URL: https://doi.org/10.1109/TIT.2012.2204032.
  11. Frank Kelly. Private communication, 2018. Google Scholar
  12. Frank Kelly and Elena Yudovina. Stochastic Networks, volume 2 of Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge, 2014. URL: https://doi.org/10.1017/CBO9781139565363.
  13. Frank P. Kelly. Stochastic models of computer communication systems. Journal of the Royal Statistical Society, 47(3):379-395, 415-428, 1985. URL: https://arxiv.org/abs/2345773.
  14. David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, USA, second edition, 2017. URL: https://doi.org/10.1090/mbk/107.
  15. Pinyan Lu, Kuan Yang, and Chihao Zhang. Fptas for hardcore and ising models on hypergraphs. In 33rd Symposium on Theoretical Aspects of Computer Science, volume 47 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 51, 14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  16. J. Ni and R. Srikant. Distributed csma/ca algorithms for achieving maximum throughput in wireless networks. In 2009 Information Theory and Applications Workshop, pages 250-250, 2009. URL: https://doi.org/10.1109/ITA.2009.5044953.
  17. Jian Ni, Bo Tan, and R. Srikant. Q-csma: Queue-length based csma/ca algorithms for achieving maximum throughput and low delay in wireless networks. In Proceedings of the 29th Conference on Information Communications, INFOCOM'10, pages 271-275, Piscataway, NJ, USA, 2010. IEEE Press. Google Scholar
  18. Ron Peled and Yinon Spinka. Long-range order in discrete spin systems. arXiv:2010.03177 [math-ph], January 2024. URL: https://doi.org/10.48550/arXiv.2010.03177.
  19. Shreevatsa Rajagopalan, Devavrat Shah, and Jinwoo Shin. Network adiabatic theorem: An efficient randomized protocol for contention resolution. SIGMETRICS Perform. Eval. Rev., 37(1):133-144, June 2009. URL: https://doi.org/10.1145/2492101.1555365.
  20. L. K. Runnels and J. L. Lebowitz. Analyticity of a hard-core multicomponent lattice gas. Journal of Statistical Physics, 14(6):525-533, 1976. URL: https://doi.org/10.1007/BF01012851.
  21. Pol Torres Compta, Frank H. P. Fitzek, and Daniel E. Lucani. Network coding is the 5g key enabling technology: Effects and strategies to manage heterogeneous packet lengths. Transactions on Emerging Telecommunications Technologies, 26(1):46-55, January 2015. URL: https://doi.org/10.1002/ett.2899.
  22. Nigel Walker. Design challenges in BT’s network. www.newton.ac.uk/seminar/23532, 2018.
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