Possibilities and Impossibilities for Distributed Subgraph Detection

In the distributed subgraph detection problem, we are given a fixed subgraph H, and the network must decide whether the network graph contains a copy of H or not. Subgraph detection can be solved in a constant number of rounds if message size is unbounded, but in the CONGEST model, where each message has bounded size, it can have high round complexity. Distributed subgraph detection has received significant attention recently, with new upper and lower bounds, but several fundamental questions remain open. In this paper we prove new possibility and impossibility results for subgraph detection in the CONGEST model. We show for the first time that some subgraphs require superlinear --- in fact, nearly quadratic --- running time, even in small-diameter networks. We also study cycle-detection, and show that any even cycle can be detected in sublinear time (in contrast to odd cycles, which require linear time). For the special case of triangle-detection, we show that deterministic algorithms require $Ømega(łog n)$ total communication even in graphs of degree 2, and that one-round randomized algorithms must send $Ømega(Δ)$ bits in graphs of degree Δ, improving on the recent results of [Abboud et. al.]. Finally, we extend a recent lower bound of [Izumi, Le Gall] on listing all triangles to cliques of any size.