License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ITCS.2022.111
URN: urn:nbn:de:0030-drops-157077
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Roghani, Mohammad ; Saberi, Amin ; Wajc, David

Beating the Folklore Algorithm for Dynamic Matching

LIPIcs-ITCS-2022-111.pdf (0.8 MB)


The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence 2-approximate) matching in O(n) worst-case update time in n-node graphs.

We present the first deterministic algorithm which outperforms the folklore algorithm in terms of both approximation ratio and worst-case update time. Specifically, we give a (2-Ω(1))-approximate algorithm with O(m^{3/8}) = O(n^{3/4}) worst-case update time in n-node, m-edge graphs. For sufficiently small constant ε > 0, no deterministic (2+ε)-approximate algorithm with worst-case update time O(n^{0.99}) was known. Our second result is the first deterministic (2+ε)-approximate weighted matching algorithm with O_ε(1)⋅ O(∜{m}) = O_ε(1)⋅ O(√n) worst-case update time. Neither of our results were previously known to be achievable by a randomized algorithm against an adaptive adversary.

Our main technical contributions are threefold: first, we characterize the tight cases for kernels, which are the well-studied matching sparsifiers underlying much of the (2+ε)-approximate dynamic matching literature. This characterization, together with multiple ideas - old and new - underlies our result for breaking the approximation barrier of 2. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the recourse of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural 3/2 factor in the approximation ratio which such approaches naturally incur (reminiscent of the integrality gap of the fractional matching polytope in general graphs).

BibTeX - Entry

  author =	{Roghani, Mohammad and Saberi, Amin and Wajc, David},
  title =	{{Beating the Folklore Algorithm for Dynamic Matching}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{111:1--111:23},
  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 =		{},
  URN =		{urn:nbn:de:0030-drops-157077},
  doi =		{10.4230/LIPIcs.ITCS.2022.111},
  annote =	{Keywords: dynamic matching, dynamic graph algorithms, sublinear algorithms}

Keywords: dynamic matching, dynamic graph algorithms, sublinear algorithms
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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