The Complexity of Symmetry Breaking in Massive Graphs

Abstract: The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related $\beta$-ruling set problem, in two computational models suited for large-scale graph processing, namely the $k$-machine model and the graph streaming model. We present a number of results. For MIS in the $k$-machine model, we improve the $\tilde{O}(m/k² + \Delta/k)$-round upper bound of Klauck et al. (SODA 2015) by presenting an $\tilde{O}(m/k<sup>2)$-round algorithm. We also present an $\tilde{\Omega}(n/k<sup>2)$ round lower bound for MIS, the first lower bound for a symmetry breaking problem in the $k$-machine model. For $\beta$-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the $k$-machine model and also in the graph streaming model. More specifically, we obtain a $k$-machine algorithm that runs in $\tilde{O}(\beta n\Delta<sup>{1/\beta}/k<sup>2)$ rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use $O(\beta \cdot n<sup>{1+1/2<sup>{\beta-1}})$ space. The latter result establishes a clear separation between MIS, which is known to require $\Omega(n<sup>2)$ space (Cormode et al., ICALP 2019), and $\beta$-ruling sets, even for $\beta = 2$. Finally, we present an even faster 2-ruling set algorithm in the $k$-machine model, one that runs in $$\tilde{O}(n/k^{2-\epsilon} + k<sup>{1-\epsilon})$$ rounds for any $\epsilon$, $0 \le \epsilon \le 1$.