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Matching Algorithms in the Sparse Stochastic Block Model

Authors Anna Brandenberger , Byron Chin , Nathan S. Sheffield , Divya Shyamal

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Author Details

Anna Brandenberger
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Byron Chin
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Nathan S. Sheffield
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Divya Shyamal
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

We thank Elchanan Mossel for suggesting the problem, and David Jerison for helpful comments.

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Anna Brandenberger, Byron Chin, Nathan S. Sheffield, and Divya Shyamal. Matching Algorithms in the Sparse Stochastic Block Model. 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. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.16

Abstract

In sparse Erdős-Rényi graphs, it is known that a linear-time algorithm of Karp and Sipser achieves near-optimal matching sizes asymptotically almost surely, giving a law-of-large numbers for the matching numbers of such graphs in terms of solutions to an ODE [Jonathan Aronson et al., 1998]. We provide an extension of this analysis, identifying broad ranges of stochastic block model parameters for which the Karp-Sipser algorithm achieves near-optimal matching sizes, but demonstrating that it cannot perform optimally on general stochastic block model instances. We also consider the problem of constructing a matching online, in which the vertices of one half of a bipartite stochastic block model arrive one-at-a-time, and must be matched as they arrive. We show that, when the expected degrees in all communities are equal, the competitive ratio lower bound of 0.837 found by Mastin and Jaillet for the Erdős-Rényi case [Andrew Mastin and Patrick Jaillet, 2013] is achieved by a simple greedy algorithm, and this competitive ratio is optimal. We then propose and analyze a linear-time online matching algorithm with better performance in general stochastic block models.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Matching Algorithms
  • Online Matching
  • Stochastic Block Model

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References

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