Near-Optimal Distributed Dominating Set in Bounded Arboricity Graphs

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, Moscibroda, and Wattenhofer JACM'16]. 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 and Wattenhofer DISC'10], a deterministic $O(\alpha \log \Delta)$ approximation in $O(\log \Delta)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(\alpha)$ approximation in $O(\log^2 \Delta)$ rounds [implicit in Bansal and Umboh IPL'17 and Kuhn, Moscibroda, and Wattenhofer SODA'06], and a randomized $O(\alpha)$ approximation in $O(\alpha\log n)$ rounds [Morgan, Solomon and Wein DISC'21]. 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 and Umboh IPL'17].