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Graphical Balanced Allocations with Removals

Authors Sam Olesker-Taylor , Thomas Sauerwald , Luca Zanetti

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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Markov processes
Keywords
  • balanced allocations
  • load balancing
  • queueing networks
  • balls-into-bins
  • Markov chains

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Abstract

We study balanced allocations on graphs with removals. Load arrives at each edge e at an exponential rate and is then allocated to the vertex incident to e with the lowest current load. Load is removed from each vertex at an exponential rate. We identify a "conductance-like" quantity that determines if an equilibrium exists and allows us to bound the maximal load at equilibrium. Our analysis, based on simple potential function arguments, is very robust and can also handle noise in how the load is allocated.
We also apply our general techniques to study the synchronous version of the process above, in which allocations and removals happen simultaneously at discrete time steps. We prove that, for any regular graph, in equilibrium, the expected difference in load across an edge, averaged over all edges, is at most 2. This implies, for example, that the two-choice process on the cycle has an O(n) gap between maximal and minimal load, improving the state-of-the-art by a log n factor.

Cite As Get BibTex

Sam Olesker-Taylor, Thomas Sauerwald, and Luca Zanetti. Graphical Balanced Allocations with Removals. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.26

Author Details

Sam Olesker-Taylor
  • Department of Statistics, University of Warwick, UK
Thomas Sauerwald
  • Department of Computer Science & Technology, University of Cambridge, UK
Luca Zanetti
  • Department of Mathematical Sciences, University of Bath, UK

Funding

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/Y032691/1]. The authors also acknowledge the Institute for Mathematical Innovation (IMI) at the University of Bath and the Centre for Research in Statistical Methodology (CRiSM) at the University of Warwick for supporting the workshop Mixing Times and Applications where part of this work originated.

References

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