License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.94
URN: urn:nbn:de:0030-drops-156909
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15690/
Go to the corresponding LIPIcs Volume Portal

Kiss, Peter

Deterministic Dynamic Matching in Worst-Case Update Time

pdf-format:
LIPIcs-ITCS-2022-94.pdf (0.8 MB)

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).

BibTeX - Entry

@InProceedings{kiss:LIPIcs.ITCS.2022.94,
  author =	{Kiss, Peter},
  title =	{{Deterministic Dynamic Matching in Worst-Case Update Time}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{94:1--94:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15690},
  URN =		{urn:nbn:de:0030-drops-156909},
  doi =		{10.4230/LIPIcs.ITCS.2022.94},
  annote =	{Keywords: Dynamic Algorithms, Matching, Approximate Matching, EDCS}
}

Keywords: Dynamic Algorithms, Matching, Approximate Matching, EDCS
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI