Distributed Triangle Detection is Hard in Few Rounds

In the distributed triangle detection problem, we have an $n$-vertex network $G=(V,E)$ with one player for each vertex of the graph who sees the edges incident on the vertex. The players communicate in synchronous rounds using the edges of this network and have a limited bandwidth of $O(\log{n})$ bits over each edge. The goal is to detect whether or not $G$ contains a triangle as a subgraph in a minimal number of rounds. We prove that any protocol (deterministic or randomized) for distributed triangle detection requires $Ω(\log\log{n})$ rounds of communication. Prior to our work, only one-round lower bounds were known for this problem. The primary technique for proving these types of distributed lower bounds is via reductions from two-party communication complexity. However, it has been known for a while that this approach is provably incapable of establishing any meaningful lower bounds for distributed triangle detection. Our main technical contribution is a new information theoretic argument which combines recent advances on multi-pass graph streaming lower bounds with the point-to-point communication aspects of distributed models, and can be of independent interest.