Near-optimal distributed dominating set in bounded arboricity graphs

Abstract

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