Distributed Lower Bounds for Ruling Sets

Given a graph $G=(V,E)$, an $(\alpha,\beta)$-ruling set is a subset $S\subseteq V$ such that the distance between any two vertices in $S$ is at least $\alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is at most $\beta$. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a $(2,\beta)$-ruling set (and hence also any $(\alpha,\beta)$-ruling set with $\alpha >2$) in the LOCAL model of distributed computing, we show the following, where $n$ denotes the number of vertices, $\Delta$ the maximum degree, and $c$ is some universal constant independent of $n$ and $\Delta$: (a) Any deterministic algorithm requires $\Omega(\min\{ \tfrac{\log\Delta}{\beta\log\log\Delta},\log_\Delta n\})$ rounds for all $\beta\le c \cdot\min\{\sqrt{\tfrac{\log\Delta}{\log\log\Delta}},\log_\Delta n\}$. By optimizing $\Delta$, this implies a deterministic lower bound of $\Omega(\sqrt{\frac{\log n}{\beta\log\log n}})$ for all $\beta\le c\,\sqrt[3]{\tfrac{\log n}{\log\log n}}$. (b) Any randomized algorithm requires $\Omega(\min\{\frac{\log\Delta}{\beta\log\log\Delta},\log_\Delta\log n\})$ rounds for all $\beta\le c\cdot\min\{\sqrt{\tfrac{\log\Delta}{\log\log\Delta}},\log_\Delta\log n\}$. By optimizing $\Delta$, this implies a randomized lower bound of $\Omega(\sqrt{\tfrac{\log \log n}{\beta\log\log\log n}})$ for all $\beta\le c\,\sqrt[3]{\tfrac{\log\log n}{\log\log\log n}}$. For $\beta > 1$, this improves on the previously best lower bound of $\Omega(\log^* n)$ rounds that follows from the 30-year-old bounds of Linial [SIAM J. Comput., 21(1992), pp. 193--201] and Naor [SIAM J. Discrete Math., 4(1991), pp. 409--412] (resp., $\Omega(1)$ rounds if $\beta \in \omega(\log^* n)$). For $\beta = 1$, i.e., for the problem of computing a maximal independent set (which is nothing else than a $(2,1)$-ruling set), our results improve on the previously best lower bound of $\Omega(\log^* n)$ on trees, as our bounds already hold on trees. For the maximal independent set on general graphs, a deterministic lower bound of $\Omega(\min\{\Delta, \log_{\Delta} n\})$ and a randomized lower bound of $\Omega(\min\{\Delta, \log_{\Delta} \log n\})$ were already known due to Balliu et al. [Proceedings of FOCS, 2019, pp. 481--497].