Distributed Dominating Set With Optimal Rounds and Message Size in Bounded Arboricity Graphs

Abstract

We study the distributed minimum dominating set problem on graphs of arboricity $\alpha$. Dory, Ghaffari, and Ilchi [PODC'22] showed that any algorithm achieving a constant or poly-logarithmic approximation factor needs at least $\Omega(\log\Delta/\log\log\Delta)$ rounds in graphs of maximum degree $\Delta$ and arboricity $\alpha$, even when $\alpha=2$ and even when the message sizes are unbounded. Although there is a variety of algorithms with a near-optimal round complexity of $O(\log\Delta)$, it is natural to ask: What is the best approximation factor in the optimal round complexity of $O(\log\Delta/\log\log\Delta)$? We make progress in answering this question by describing a deterministic algorithm that obtains a $O\left( \alpha \log \Delta / \log\log \Delta \right)$ approximation without prior knowledge of $\alpha$ with optimal round complexity of $O\left( \log \Delta / \log\log \Delta \right)$ 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 $O\left( \log \Delta / \log\log \Delta \right)$ without prior knowledge of $\alpha$ is due to Lenzen and Wattenhofer [DISC'10] that obtains a $O(\alpha \log^{1+\varepsilon}\Delta / (\varepsilon\log\log \Delta))$ approximation in $O(\log\Delta/(\varepsilon\log\log\Delta))$ rounds and $O(\log(\varepsilon^{-1}\log\Delta))$ 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 $\Delta$. 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 $\alpha$, which is far from optimal.