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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.DISC.2021.33
URN: urn:nbn:de:0030-drops-148353
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14835/
Morgan, Adir ;
Solomon, Shay ;
Wein, Nicole
Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial
Abstract
We revisit the minimum dominating set problem on graphs with arboricity bounded by α. In the (standard) centralized setting, Bansal and Umboh [Bansal and Umboh, 2017] gave an O(α)-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [Koufogiannakis and Young, 2014; Young, 2014; Allen-Zhu and Orecchia, 2019; Quanrud, 2020]. Moreover, [Bansal and Umboh, 2017] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [Christoph Lenzen and Roger Wattenhofer, 2010], and Jones et al. [Jones et al., 2013], achieve an approximation factor of O(α²) in linear time.
There is a similar situation in the distributed setting: While there is an O(log² n)-round LP-based O(α)-approximation algorithm implied in [Kuhn et al., 2006], the best non-LP-based algorithm by Lenzen and Wattenhofer [Christoph Lenzen and Roger Wattenhofer, 2010] is an implementation of their centralized algorithm, providing an O(α²)-approximation within O(log n) rounds.
We address the questions of whether one can achieve an O(α)-approximation algorithm that is elementary, i.e., not based on any LP-based methods, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. Our contribution is two-fold:
1) In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an O(α)-approximation in linear time. The previous state-of-the-art O(α)-approximation algorithms are (1) LP-based, (2) more complicated, and (3) have super-linear running time.
2) Based on our centralized algorithm, we design a distributed combinatorial O(α)-approximation algorithm in the CONGEST model that runs in O(αlog n) rounds with high probability. Not only does this result provide the first nontrivial non-LP-based distributed o(α²)-approximation algorithm for this problem, it also outperforms the best LP-based distributed algorithm for a wide range of parameters.
BibTeX - Entry
@InProceedings{morgan_et_al:LIPIcs.DISC.2021.33,
author = {Morgan, Adir and Solomon, Shay and Wein, Nicole},
title = {{Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial}},
booktitle = {35th International Symposium on Distributed Computing (DISC 2021)},
pages = {33:1--33:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-210-5},
ISSN = {1868-8969},
year = {2021},
volume = {209},
editor = {Gilbert, Seth},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14835},
URN = {urn:nbn:de:0030-drops-148353},
doi = {10.4230/LIPIcs.DISC.2021.33},
annote = {Keywords: Graph Algorithms, Dominating Set, Bounded Arboricity, Linear time algorithms}
}
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Keywords: |
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Graph Algorithms, Dominating Set, Bounded Arboricity, Linear time algorithms |
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Collection: |
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35th International Symposium on Distributed Computing (DISC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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04.10.2021 |
