  Motivated by the serious problem that hospitals in rural areas suffer from a
shortage of residents, we study the Hospitals/Residents model in which
hospitals are associated with lower quotas and the objective is to satisfy them
as much as possible. When preference lists are strict, the number of residents
assigned to each hospital is the same in any stable matching because of the
well-known rural hospitals theorem; thus there is no room for algorithmic
interventions. However, when ties are introduced to preference lists, this will
no longer apply because the number of residents may vary over stable matchings.
In this paper, we formulate an optimization problem to find a stable matching
with the maximum total satisfaction ratio for lower quotas. We first
investigate how the total satisfaction ratio varies over choices of stable
matchings in four natural scenarios and provide the exact values of these
maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm
for our problem; in one scenario it solves the problem optimally, and in the
other three scenarios, which are NP-hard, it yields a better approximation
factor than that of a naive tie-breaking method. Finally, we show
inapproximability results for the above-mentioned three NP-hard scenarios.
