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Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial

Authors Adir Morgan, Shay Solomon, Nicole Wein

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Algorithms
  • Dominating Set
  • Bounded Arboricity
  • Linear time algorithms

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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.

Cite As Get BibTex

Adir Morgan, Shay Solomon, and Nicole Wein. Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.DISC.2021.33

Author Details

Adir Morgan
  • Tel Aviv University, Israel
Shay Solomon
  • Tel Aviv University, Israel
Nicole Wein
  • MIT, Cambridge, MA, USA

Funding

  • Solomon, Shay: Partially supported by the Israel Science Foundation grant No.1991/19.
  • Wein, Nicole: Supported by NSF Grant CCF-1514339.

Acknowledgements

We thank Yosi Hezi and Quanquan Liu for fruitful discussions. We would also like to thank Neal E. Young and Kent Quanrud for correspondence about their prior work.

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