Phase-Bounded Broadcast Networks over Topologies of Communication

We study networks of processes that all execute the same finite state protocol and that communicate through broadcasts. The processes are organized in a graph (a topology) and only the neighbors of a process in this graph can receive its broadcasts. The coverability problem asks, given a protocol and a state of the protocol, whether there is a topology for the processes such that one of them (at least) reaches the given state. This problem is undecidable. We study here an under-approximation of the problem where processes alternate a bounded number of times $k$ between phases of broadcasting and phases of receiving messages. We show that, if the problem remains undecidable when $k$ is greater than 6, it becomes decidable for $k=2$, and EXPSPACE-complete for $k=1$. Furthermore, we show that if we restrict ourselves to line topologies, the problem is in $P$ for $k=1$ and $k=2$.