Near-optimal distributed dominating set in bounded arboricity graphs

We describe a simple deterministic O(ε^-1logΔ) round distributed algorithm for (2α+1)(1+ε) approximation of minimum weighted dominating set on graphs with arboricity at most α. Here Δ 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(α²) approximation in O(logn) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic O(αlogΔ) approximation in O(logΔ) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic O(α) approximation in O(log²Δ) 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(α) approximation in O(αlogn) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized O(αlogΔ) round distributed algorithm that sharpens the approximation factor to α(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 α-1-ε approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).