We describe a simple deterministic \(O( \varepsilon ^{-1} \log \Delta )\) round distributed algorithm for \((2\alpha +1)(1 + \varepsilon )\) approximation of minimum weighted dominating set on graphs with arboricity at most \(\alpha \). Here \(\Delta \) denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized \(O(\alpha ^2)\) approximation in \(O(\log n)\) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic \(O(\alpha \log \Delta )\) approximation in \(O(\log \Delta )\) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic \(O(\alpha )\) approximation in \(O(\log ^2 \Delta )\) rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized \(O(\alpha )\) approximation in \(O(\alpha \log n)\) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized \(O(\alpha \log \Delta )\) round distributed algorithm that sharpens the approximation factor to \(\alpha (1+o(1))\). If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve \(\alpha - 1 - \varepsilon \) approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).

我们描述了一个简单的确定性\(O( \varepsilon ^{-1} \log \Delta )\)轮分布算法，用于\((2\alpha +1)(1 + \varepsilon )\)最小加权支配集的近似在最多\(\alpha \)的图上。这里\(\Delta \)表示最大度数。我们还展示了一个下限，通过减少分布式顶点覆盖近似的著名 KMW 下限（Kuhn et al. in JACM 63:116, 2016），即使对于未加权的情况，这一轮复杂度也几乎是最优的。我们的算法改进了所有先前的结果（仅适用于未加权的图），包括\(O(\log n)\)中的随机\(O(\alpha ^2)\)近似rounds（Lenzen et al. in International symposium on distributed computing, Springer, 2010），在\(O(\log \Delta )\) rounds 中的确定性 \ (O(\alpha \log \Delta )\ )近似（Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic \(O(\alpha )\) approximation in \(O(\log ^2 \Delta )\) rounds（隐含在 Bansal 等人的 Inform 中Process Lett 122:21–24, 2017；第 17 届离散算法研讨会 (SODA)，2006 年)，以及 \ (O(\alpha \log n)\)轮中的随机\(O(\alpha )\)近似（Morgan 等人在第 35 届国际分布式计算研讨会上，2021 年）。我们还提供随机\(O(\alpha \log \Delta )\)圆形分布式算法，将近似因子锐化为\(\alpha (1+o(1))\)。如果每个节点都被限制进行多项式时间计算，我们的近似因子在一阶是紧的，因为它很难实现 \( \alpha - 1 - \varepsilon \)近似（Bansal 等人在 Inform Process Lett 中122:21-24, 2017)。