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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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