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.
@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/entities/document/10.4230/LIPIcs.DISC.2021.33}, 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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