We study the distributed minimum dominating set problem on graphs of arboricity $α$. Dory, Ghaffari, and Ilchi [PODC'22] showed that any algorithm achieving a constant or poly-logarithmic approximation factor needs at least $Ω(\logΔ/\log\logΔ)$ rounds in graphs of maximum degree $Δ$ and arboricity $α$, even when $α=2$ and even when the message sizes are unbounded. Although there is a variety of algorithms with a near-optimal round complexity of $O(\logΔ)$, it is natural to ask: What is the best approximation factor in the optimal round complexity of $O(\logΔ/\log\logΔ)$? We make progress in answering this question by describing a deterministic algorithm that obtains a $O\left( α\log Δ/ \log\log Δ\right)$ approximation without prior knowledge of $α$ with optimal round complexity of $O\left( \log Δ/ \log\log Δ\right)$ and optimal message size of $1$ bit per round. Among all of the previous results, the only algorithm that achieves the optimal round complexity of $O\left( \log Δ/ \log\log Δ\right)$ without prior knowledge of $α$ is due to Lenzen and Wattenhofer [DISC'10] that obtains a $O(α\log^{1+\varepsilon}Δ/ (\varepsilon\log\log Δ))$ approximation in $O(\logΔ/(\varepsilon\log\logΔ))$ rounds and $O(\log(\varepsilon^{-1}\logΔ))$ message size. Our algorithm simplifies and improves upon this result. The only downside of our algorithm compared to the algorithm of Lenzen and Wattenhofer is that it needs prior knowledge of $Δ$. The previous state-of-the-art algorithm by Dory, Ghaffari, and Ilchi [PODC'22] has a dependency on $\log n$ in the round complexity for unknown $α$, which is far from optimal.

翻译：我们研究树状度为 $α$ 的图上的分布式最小支配集问题。Dory、Ghaffari 和 Ilchi [PODC'22] 表明，即使在 $α=2$ 且消息大小无限制的情况下，任何实现常数或多对数近似因子的算法在最大度为 $Δ$、树状度为 $α$ 的图中至少需要 $Ω(\logΔ/\log\logΔ)$ 轮。尽管存在多种轮复杂度接近最优 $O(\logΔ)$ 的算法，但自然产生一个问题：在最优轮复杂度 $O(\logΔ/\log\logΔ)$ 下，最佳近似因子是多少？我们通过描述一种确定性算法来推进对该问题的解答，该算法在无需预先知道 $α$ 的情况下，以最优轮复杂度 $O\left( \log Δ/ \log\log Δ\right)$ 和每轮最优消息大小 $1$ 比特，获得 $O\left( α\log Δ/ \log\log Δ\right)$ 近似。在所有先前结果中，唯一能在无需预先知道 $α$ 的情况下达到最优轮复杂度 $O\left( \log Δ/ \log\log Δ\right)$ 的算法来自 Lenzen 和 Wattenhofer [DISC'10]，该算法在 $O(\logΔ/(\varepsilon\log\logΔ))$ 轮和 $O(\log(\varepsilon^{-1}\logΔ))$ 消息大小下获得 $O(α\log^{1+\varepsilon}Δ/ (\varepsilon\log\log Δ))$ 近似。我们的算法简化并改进了这一结果。相对于 Lenzen 和 Wattenhofer 的算法，我们算法的唯一缺点是需要预先知道 $Δ$。Dory、Ghaffari 和 Ilchi [PODC'22] 提出的先前最先进算法在未知 $α$ 的情况下，轮复杂度依赖于 $\log n$，这远非最优。