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

We revisit the minimum dominating set problem on graphs with arboricity $\alpha$. In the (standard) centralized setting, Bansal and Umboh [BU17] gave an $O(\alpha)$-approximation LP rounding algorithm, which translates into a near-linear time algorithm using general-purpose approximation results for explicit 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. [JLR+13], achieve an approximation factor of $O(\alpha^2)$ in linear time. There is a similar situation in the distributed setting: While there is an $O(\log^2 n)$-round LP-based $O(\alpha)$-approximation algorithms [KMW06, DKM19], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(\alpha^2)$-approximation within $O(\log n)$ rounds with high probability. We address the questions of whether one can achieve an $O(\alpha)$-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: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and runtime: an $O(\alpha)$-approximation in linear time. The previous best $O(\alpha)$-approximation algorithms are LP-based and have super-linear running time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(\alpha)$-approximation algorithm in the $\mathsf{CONGEST}$ model that runs in $O(\alpha\log n)$ rounds with high probability.