  We study a simple exchange market, introduced by Gourv\'{e}s, Lesca and
Wilczynski (IJCAI-17), where every agent initially holds a single object. The
agents have preferences over the objects, and two agents may swap their objects
if they both prefer the object of the other agent. The agents live in an
underlying social network that governs the structure of the swaps: Two agents
can only swap their objects if they are adjacent. We investigate the REACHABLE
OBJECT problem, which asks whether a given starting situation can ever lead, by
means of a sequence of swaps, to a situation where a given agent obtains a
given object. Our results answer several central open questions on the
complexity of REACHABLE OBJECT. First, the problem is polynomial-time solvable
if the social network is a path. Second, the problem is NP-hard on cliques and
generalized caterpillars. Finally, we establish a three-versus-four dichotomy
result for preference lists of bounded length: The problem is easy if all
preference lists have length at most three, and the problem becomes NP-hard
even if all agents have preference lists of length at most four.
