Finding Cut-Offs in Leaderless Rendez-Vous Protocols is Easy

In rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number $B$ such that all initial configurations of the protocol with at least $B$ agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper (Horn and Sangnier, CONCUR 2020), Horn and Sangnier prove that the cut-off problem is equivalent to the Petri net reachability problem for protocols with a leader, and in EXPSPACE for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to PSPACE and NP, respectively. The problem of lowering these upper bounds or finding matching lower bounds is left open. We show that the cut-off problem is P-complete for leaderless protocols, NP-complete for symmetric protocols with a leader, and in NC for leaderless symmetric protocols, thereby solving all the problems left open in (Horn and Sangnier, CONCUR 2020). Further, we also consider a variant of the cut-off problem suggested in (Horn and Sangnier, CONCUR 2020) and prove that that variant is P-complete for leaderless protocols and NL-complete for leaderless symmetric protocols.