  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 proved that the cut-off problem is decidable (and at
least as hard as 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 was left open. We show that the cut-off problem is P-complete for
leaderless protocols and in NC for leaderless symmetric protocols. Further, we
also consider a variant of the cut-off problem suggested in (Horn and Sangnier,
CONCUR 2020), which we call the bounded-loss cut-off problem and prove that
this problem is P-complete for leaderless protocols and NL-complete for
leaderless symmetric protocols. Finally, by reusing some of the techniques
applied for the analysis of leaderless protocols, we show that the cut-off
problem for symmetric protocols with a leader is NP-complete, thereby improving
upon all the elementary upper bounds of (Horn and Sangnier, CONCUR 2020).
