The Complexity of Symmetry Breaking in Massive Graphs

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related β-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 Õ(m/k² + ∆/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an Õ(m/k²)-round algorithm. We also present an Ω˜(n/k²) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For β-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 Õ(βn∆^1/β/k²) 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(β · n^1+1/2β−1) space. The latter result establishes a clear separation between MIS, which is known to require Ω(n²) space (Cormode et al., ICALP 2019), and β-ruling sets, even for β = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in Õ(n/k^2−ε + k^1−ε) rounds for any ε, 0 ≤ ε ≤ 1. For a wide range of values of k this round complexity simplifies to Õ(n/k²) rounds, which we conjecture is optimal. 
Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L₀-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems.