TY - GEN
T1 - Algorithms for the minimum dominating set problem in bounded arboricity graphs
T2 - 35th International Symposium on Distributed Computing, DISC 2021
AU - Morgan, Adir
AU - Solomon, Shay
AU - Wein, Nicole
N1 - Publisher Copyright:
© Adir Morgan, Shay Solomon, and Nicole Wein; licensed under Creative Commons License CC-BY 4.0
PY - 2021/10/1
Y1 - 2021/10/1
N2 - 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.
AB - 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.
KW - Bounded Arboricity
KW - Dominating Set
KW - Graph Algorithms
KW - Linear time algorithms
UR - http://www.scopus.com/inward/record.url?scp=85118147046&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.DISC.2021.33
DO - 10.4230/LIPIcs.DISC.2021.33
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AN - SCOPUS:85118147046
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 35th International Symposium on Distributed Computing, DISC 2021
A2 - Gilbert, Seth
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Y2 - 4 October 2021 through 8 October 2021
ER -