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).

翻译：在会合协议中，任意大量无法区分的有限状态代理以成对方式交互。截止值问题询问是否存在一个数 $B$，使得协议中所有初始配置（在给定初始状态下至少有 $B$ 个代理）都能达到所有代理处于给定最终状态的最终配置。在近期论文中（Horn 和 Sangnier, CONCUR 2020），Horn 和 Sangnier 证明了对于有领导者的协议，截止值问题可判定（且至少与 Petri 网可达性问题难度相当），而对于无领导者协议，该问题属于 EXPSPACE 复杂度类。此外，对于对称协议这一特殊类别，他们分别将上述界限降低至 PSPACE 和 NP。降低这些上界或寻找匹配下界的问题当时尚未解决。我们证明：无领导者协议的截止值问题是 P-完全的，而无领导者对称协议的此类问题属于 NC 复杂度类。此外，我们考虑了 (Horn 和 Sangnier, CONCUR 2020) 中提出的一种截止值问题的变体，称之为有界损失截止值问题，并证明该问题对无领导者协议是 P-完全的，对无领导者对称协议是 NL-完全的。最后，通过复用部分用于分析无领导者协议的技术，我们证明有领导者对称协议的截止值问题是 NP-完全的，从而改进了 (Horn 和 Sangnier, CONCUR 2020) 中所有基本上界。