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 [BU17] 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 [KY14, You14, AZO19, Qua20]. Moreover, [BU17] 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 [LW10], and Jones et al. [JLR13], achieve an approximation factor of O(α) in linear time. There is a similar situation in the distributed setting: While there is an O(log n)-round LP-based O(α)-approximation algorithm implied in [KMW06], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing anO(α)-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. More specifically, 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(α)-approximation algorithm for this problem, it also outperforms the best LP-based distributed algorithm for a wide range of parameters. *Email: [email protected] Partially supported by the Israel Science Foundation grant No.1991/19. Email: [email protected] Supported by NSF Grant CCF-1514339. Email: [email protected] ar X iv :2 10 2. 10 07 7v 3 [ cs .D S] 1 0 A ug 2 02 1