  The efficient and fair distribution of indivisible resources among agents is
a common problem in the field of \emph{Multi-Agent-Systems}. We consider a
graph-based version of this problem called Reachable Assignments, introduced by
Gourves, Lesca, and Wilczynski [AAAI, 2017]. The input for this problem
consists of a set of agents, a set of objects, the agent's preferences over the
objects, a graph with the agents as vertices and edges encoding which agents
can trade resources with each other, and an initial and a target distribution
of the objects, where each agent owns exactly one object in each distribution.
The question is then whether the target distribution is reachable via a
sequence of rational trades. A trade is rational when the two participating
agents are neighbors in the graph and both obtain an object they prefer over
the object they previously held. We show that Reachable Assignments is NP-hard
even when restricting the input graph to be a clique and develop an
$O(n^3)$-time algorithm for the case where the input graph is a cycle with $n$
vertices.
