  The \textsc{Housing Market} problem is a widely studied resource allocation
problem. In this problem, each agent can only receive a single object and has
preferences over all objects. Starting from an initial endowment, we want to
reach a certain assignment via a sequence of rational trades. We first consider
whether an object is reachable for a given agent under a social network, where
a trade between two agents is allowed if they are neighbors in the network and
no participant has a deficit from the trade. Assume that the preferences of the
agents are strict (no tie among objects is allowed). This problem is
polynomially solvable in a star-network and NP-complete in a tree-network. It
is left as a challenging open problem whether the problem is polynomially
solvable when the network is a path. We answer this open problem positively by
giving a polynomial-time algorithm. Then we show that when the preferences of
the agents are weak (ties among objects are allowed), the problem becomes
NP-hard even when the network is a path. In addition, we consider the
computational complexity of finding different optimal assignments for the
problem under the network being a path or a star.
