Detecting cliques in CONGEST networks

The problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph $H$, the input network contains a subgraph isomorphic to $H$ or not. We investigate this problem for $H$ being a clique $K_{l}$ in the classical distributed CONGEST model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links. Our first and main result is a lower bound, showing that detecting $K_{l}$ requires $Ω(\sqrt{n} / b)$ communication rounds, for every $4 \le l \le \sqrt{n}$, and $Ω(n / (l b))$ rounds for every $l \ge \sqrt{n}$, where $b$ is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for $K_4$ detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework.