Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Khanna, Sanjeev; Konrad, Christian https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-156894
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Optimal Bounds for Dominating Set in Graph Streams

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Abstract

We resolve the space complexity of one-pass streaming algorithms for Minimum Dominating Set (MDS) in both insertion-only and insertion-deletion streams (up to poly-logarithmic factors) where an input graph is revealed by a sequence of edge updates. Recently, streaming algorithms for the related Set Cover problem have received significant attention. Even though MDS can be viewed as a special case of Set Cover, it is however harder to solve in the streaming setting since the input stream consists of individual edges rather than entire vertex-neighborhoods, as is the case in Set Cover.
We prove the following results (n is the number of vertices of the input graph):
1) In insertion-only streams, we give a one-pass semi-streaming algorithm (meaning Õ(n) space) with approximation factor Õ(√n). We also prove that every one-pass streaming algorithm with space o(n) has an approximation factor of Ω(n/log n).
Combined with a result by [Assadi et al., STOC'16] for Set Cover which, translated to MDS, shows that space Θ̃(n² / α) is necessary and sufficient for computing an α-approximation for every α = o(√n), this completely settles the space requirements for MDS in the insertion-only setting.
2) In insertion-deletion streams, we prove that space Ω(n² / (α log n)) is necessary for every approximation factor α ≤ Θ(n / log³ n). Combined with the Set Cover algorithm of [Assadi et al., STOC'16], which can be adapted to MDS even in the insertion-deletion setting to give an α-approximation in Õ(n² / α) space, this completely settles the space requirements for MDS in the insertion-deletion setting.

BibTeX - Entry

@InProceedings{khanna_et_al:LIPIcs.ITCS.2022.93,
  author =	{Khanna, Sanjeev and Konrad, Christian},
  title =	{{Optimal Bounds for Dominating Set in Graph Streams}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{93:1--93:23},
  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 =		{https://drops.dagstuhl.de/opus/volltexte/2022/15689},
  URN =		{urn:nbn:de:0030-drops-156894},
  doi =		{10.4230/LIPIcs.ITCS.2022.93},
  annote =	{Keywords: Streaming algorithms, communication complexity, information complexity, dominating set}
}

Keywords: Streaming algorithms, communication complexity, information complexity, dominating set
Seminar: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue date: 2022
Date of publication: 25.01.2022


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