  In recent work, Gourv\`es, Lesca, and Wilczynski propose a variant of the
classic housing markets model where the matching between agents and objects
evolves through Pareto-improving swaps between pairs of adjacent agents in a
social network. To explore the swap dynamics of their model, they pose several
basic questions concerning the set of reachable matchings. In their work and
other follow-up works, these questions have been studied for various classes of
graphs: stars, paths, generalized stars (i.e., trees where at most one vertex
has degree greater than two), trees, and cliques. For generalized stars and
trees, it remains open whether a Pareto-efficient reachable matching can be
found in polynomial time.
  In this paper, we pursue the same set of questions under a natural variant of
their model. In our model, the social network is replaced by a network of
objects, and a swap is allowed to take place between two agents if it is
Pareto-improving and the associated objects are adjacent in the network. In
those cases where the question of polynomial-time solvability versus
NP-hardness has been resolved for the social network model, we are able to show
that the same result holds for the network-of-objects model. In addition, for
our model, we present a polynomial-time algorithm for computing a
Pareto-efficient reachable matching in generalized star networks. Moreover, the
object reachability algorithm that we present for path networks is
significantly faster than the known polynomial-time algorithms for the same
question in the social network model.
