Reachable Globally Pareto-Efficient Allocations on Trees Are NP-Complete

We settle the complexity of reaching a globally Pareto-efficient allocation by mutually improving swaps along the edges of a tree. The problem is NP-complete, even though reachability of a specified full allocation is polynomial-time decidable on trees. The proof is a linear-size reduction from the top-target restriction of object reachability on trees. Its two devices are a fixed sentinel, which certifies global Pareto inefficiency whenever the target object has not arrived, and private cleanup leaves, which turn every successful target-reaching state into a globally Pareto-efficient allocation. The construction doubles the number of agents, preserves the tree property, and increases maximum degree by at most one. We also separate the standard global notion used here from Pareto maximality within the reachable set, which is the notion used in the earlier social-network allocation literature.