  We revisit the minimum dominating set problem on graphs with arboricity
bounded by $\alpha$. In the (standard) centralized setting, Bansal and Umboh
[BU17] gave an $O(\alpha)$-approximation LP rounding algorithm. Moreover,
[BU17] showed that it is NP-hard to achieve an asymptotic improvement. 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 are
$\text{poly}\log n$-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 question of whether one can achieve a simple, elementary
$O(\alpha)$-approximation algorithm not based on any LP-based methods, either
in the centralized setting or in the distributed setting. We resolve these
questions in the affirmative. 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 $\mathsf{CONGEST}$ model that runs
in $O(\alpha\log n )$ rounds with high probability. Our round complexity
outperforms the best LP-based distributed algorithm for a wide range of
parameters.
