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

TL;DR

This paper investigates the computational complexity of the cut-off problem in leaderless rendez-vous protocols, providing new tight bounds and extending results to symmetric and leader-based variants.

Contribution

It establishes that the cut-off problem is P-complete for leaderless protocols, in NC for symmetric cases, and NP-complete for protocols with a leader, improving previous bounds.

Findings

Cut-off problem is P-complete for leaderless protocols.

In NC for leaderless symmetric protocols.

NP-complete for symmetric protocols with a leader.

Abstract

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).