Near-Optimal Distributed 2-Ruling Sets on Graphs with Low Arboricity

Abstract

Given a graph $G=(V,E)$, a $\beta$-ruling set is a subset of nodes $S\subseteq V$ that is independent, and each node in $V$ is at distance at most $\beta$ from some node in $S$. In this paper, we present almost optimal distributed algorithms for finding $2$-ruling sets in the classical \LOCAL model. Our main contribution is a randomized algorithm that w.h.p.\ computes a $2$-ruling set on any $n$-node graph with bounded arboricity in $O(\log \log n)$ rounds. In fact, the algorithm works up to arboricity $O(\log\log n)$, improves exponentially over the prior state of the art that can be achieved by combining [Barenboim, Elkin, Pettie, Schneider; JACM'16], [Ghaffari; SODA'16], and [Bisht, Kothapalli and Pemmaraju; PODC'14], and nearly matches the lower bound of $\Omega(\log \log n / \log \log \log n)$ [Balliu, Brandt, Kuhn, Olivetti; FOCS'20]. The domination parameter $\beta=2$ is optimal for algorithms with runtime $\log^{o(1)}n$: on graphs with arboricity $2$, there is a lower bound of $\Omega(\sqrt{\log n})$ rounds for MIS (i.e., $\beta = 1$) [Khoury, Schild; FOCS'25]. Additionally, we obtain improved algorithms for larger arboricity. For general graphs with arboricity $\alpha$, we present a randomized algorithm that computes a $2$-ruling set in $\widetilde{O}(\log^{5/8} \alpha +\log^{5/3} \log n)$ rounds. This improves exponentially over the state of the art for a large range of non-constant arboricity. Our techniques extend beyond distributed computing. We present an $O(\log \log \log n)$-round algorithm in the low-space Massively Parallel Computation (\mpc) model that w.h.p.\ computes a $2$-ruling set on any graph with arboricity up to $2^{poly (\log \log n)}$, improving exponentially over the state of the art from [Kothapalli, Pai, Pemmaraju; FSTTCS'20] combined with [Fischer, Giliberti, Grunau; SPAA'23].