Near-optimal distributed dominating set in bounded arboricity graphs

We describe a simple deterministic $$ round distributed algorithm for $$ approximation of minimum weighted dominating set on graphs with arboricity at most $$ . Here $$ 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 $$ approximation in $$ rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic $$ approximation in $$ rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic $$ approximation in $$ 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 $$ approximation in $$ rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized $$ round distributed algorithm that sharpens the approximation factor to $$ . 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 $$ approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).