Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Faour, Salwa; Fuchs, Marc; Kuhn, Fabian https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

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Abstract

We provide CONGEST model algorithms for approximating the minimum weighted vertex cover and the maximum weighted matching problem. For bipartite graphs, we show that a (1+ε)-approximate weighted vertex cover can be computed deterministically in poly((log n)/ε) rounds. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least λ⋅ w(V) (where w(V) denotes the total weight of all nodes) in polylogarithmic time in the CONGEST model, one can compute a (2-2λ +ε)-approximate weighted vertex cover in poly((log n)/ε) rounds in the CONGEST model. Our result in particular implies that in graphs of arboricity a, one can compute a (2-1/a+ε)-approximate weighted vertex cover problem in poly((log n)/ε) rounds in the CONGEST model.
For maximum weighted matchings, we show that a (1-ε)-approximate solution can be computed deterministically in time 2^{O(1/ε)}⋅ polylog n in the CONGEST model. We also provide a randomized algorithm that with arbitrarily good constant probability succeeds in computing a (1-ε)-approximate weighted matching in time 2^{O(1/ε)}⋅ polylog(Δ W)⋅ log^* n, where W denotes the ratio between the largest and the smallest edge weight. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17], who gave 2^{O(1/ε)}⋅ log n and 2^{O(1/ε)}⋅ (logΔ)/(log logΔ)-round randomized approximations for the unweighted matching problem.
Finally, we show that even in the LOCAL model and in bipartite graphs of degree ≤ 3, if ε < ε₀ for some constant ε₀ > 0, then computing a (1+ε)-approximation for the unweighted minimum vertex cover problem requires Ω((log n)/ε) rounds. This generalizes a result of [Göös, Suomela; DISC '12], who showed that computing a (1+ε₀)-approximation in such graphs requires Ω(log n) rounds.

BibTeX - Entry

@InProceedings{faour_et_al:LIPIcs.OPODIS.2021.17,
  author =	{Faour, Salwa and Fuchs, Marc and Kuhn, Fabian},
  title =	{{Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings}},
  booktitle =	{25th International Conference on Principles of Distributed Systems (OPODIS 2021)},
  pages =	{17:1--17:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-219-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{217},
  editor =	{Bramas, Quentin and Gramoli, Vincent and Milani, Alessia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15792},
  URN =		{urn:nbn:de:0030-drops-157928},
  doi =		{10.4230/LIPIcs.OPODIS.2021.17},
  annote =	{Keywords: distributed graph algorithms, minimum weighted vertex cover, maximum weighted matching, distributed optimization, CONGEST model}
}

Keywords: distributed graph algorithms, minimum weighted vertex cover, maximum weighted matching, distributed optimization, CONGEST model
Seminar: 25th International Conference on Principles of Distributed Systems (OPODIS 2021)
Issue date: 2022
Date of publication: 28.02.2022


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