  We revisit the minimum dominating set problem on graphs with arboricity
bounded by $\alpha$. Bansal and Umboh [BU17] gave an $O(\alpha)$-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, Qua10]. 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 algorithm implied in
[KMW06], 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.
  We address the questions of whether one can achieve an
$O(\alpha)$-approximation algorithm that is not LP-based, 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 $O(\alpha)$-approximation in linear
time. 2. Based on our centralized algorithm, we design a distributed
combinatorial $O(\alpha)$-approximation algorithm in the CONGEST model that
runs in $O(\alpha\log n )$ rounds with high probability.
