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Deterministic Dynamic Matching in Worst-Case Update Time

Author Peter Kiss

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  • Theory of computation
Keywords
  • Dynamic Algorithms
  • Matching
  • Approximate Matching
  • EDCS

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Abstract

We present deterministic algorithms for maintaining a (3/2 + ε) and (2 + ε)-approximate maximum matching in a fully dynamic graph with worst-case update times Ô(√n) and Õ(1) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio (2 - δ) (for any δ > 0) and (2 + ε) were both shown by Roghani et al. [arXiv'2021] with update times O(n^{3/4}) and O_ε(√n) respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are O_ε(√n) and Õ(1) which were shown in Bernstein and Stein [SODA'2021] and Bhattacharya and Kiss [ICALP'2021] respectively. 
The algorithm achieving (3/2 + ε) approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein [ICALP'2015]. Say that H is a (α, δ)-approximate matching sparsifier if at all times H satisfies that μ(H) ⋅ α + δ ⋅ n ≥ μ(G) (define (α, δ)-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a (3/2 + ε, δ)-approximate matching sparsifier. We further show how to reduce the maintenance of an α-approximate maximum matching to the maintenance of an (α, δ)-approximate maximum matching building based on an observation of Assadi et al. [EC'2016]. Our reduction requires an update time blow-up of Ô(1) or Õ(1) and is deterministic or randomized against an adaptive adversary respectively. 
To achieve (2 + ε)-approximation we improve on the update time guarantee of an algorithm of Bhattacharya and Kiss [ICALP'2021]. In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak [STOC'2017] and Bernstein et al. [arXiv'2020] which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time. 
Independent Work: Independently and concurrently to our work Grandoni et al. [arXiv'2021] has presented a fully dynamic algorithm for maintaining a (3/2 + ε)-approximate maximum matching with deterministic worst-case update time O_ε(√n).

Cite As Get BibTex

Peter Kiss. Deterministic Dynamic Matching in Worst-Case Update Time. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 94:1-94:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.94

Author Details

Peter Kiss
  • Department of Computer Science, University of Warwick, Coventry, UK

Funding

  • Kiss, Peter: This work is supported by Engineering and Physical Sciences Research Council, UK (EPSRC) GrantEP/S03353X/1.

Acknowledgements

We would like to thank Sayan Bhattacharya and Thatchaphol Saranurak for helpful discussions. Further we would like to thank Thatchaphol Saranurak for suggesting to use lossless expanders to deterministically generate ε-matching preserving partitionings.

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