Algorithms for the minimum dominating set problem in bounded arboricity graphs: Simpler, faster, and combinatorial

Adir Morgan, Shay Solomon, Nicole Wein

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

We revisit the minimum dominating set problem on graphs with arboricity bounded by α. In the (standard) centralized setting, Bansal and Umboh [6] 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 [39, 57, 1, 50]. Moreover, [6] 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 [43], and Jones et al. [36], 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 [40], the best non-LP-based algorithm by Lenzen and Wattenhofer [43] 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(α log 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.

Original languageEnglish
Title of host publication35th International Symposium on Distributed Computing, DISC 2021
EditorsSeth Gilbert
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772105
DOIs
StatePublished - 1 Oct 2021
Event35th International Symposium on Distributed Computing, DISC 2021 - Virtual, Freiburg, Germany
Duration: 4 Oct 20218 Oct 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume209
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Distributed Computing, DISC 2021
Country/TerritoryGermany
CityVirtual, Freiburg
Period4/10/218/10/21

Funding

FundersFunder number
National Science FoundationCCF-1514339
Israel Science Foundation1991/19

    Keywords

    • Bounded Arboricity
    • Dominating Set
    • Graph Algorithms
    • Linear time algorithms

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