  In this paper, we consider the Hospital Residents problem (HR) and the
Hospital Residents problem with Lower Quotas (HRLQ). In this model with two
sided preferences, stability is a well accepted notion of optimality. However,
in the presence of lower quotas, a stable and feasible matching need not exist.
For the HRLQ problem, our goal therefore is to output a good feasible matching
assuming that a feasible matching exists. Computing matchings with minimum
number of blocking pairs (MinBP) and minimum number of blocking residents
(MinBR) are known to be NP-Complete. The only approximation algorithms for
these problems work under severe restrictions on the preference lists. We
present an algorithm which circumvents this restriction and computes a popular
matching in the HRLQ instance. We show that on data-sets generated using
various generators, our algorithm performs very well in terms of blocking pairs
and blocking residents. Yokoi (ISAAC 2017) recently studied envy-free matchings
for the HRLQ problem. We propose a simple modification to Yokoi's algorithm to
output a maximal envy-free matching. We observe that popular matchings
outperform envy-free matchings on several parameters of practical importance,
like size, number of blocking pairs, number of blocking residents.
  In the absence of lower quotas, that is, in the Hospital Residents (HR)
problem, stable matchings are guaranteed to exist. Even in this case, we show
that popularity is a practical alternative to stability. For instance, on
synthetic data-sets generated using a particular model, as well as on real
world data-sets, a popular matching is on an average 8-10% larger in size,
matches more number of residents to their top-choice, and more residents prefer
the popular matching as compared to a stable matching. Our comprehensive study
reveals the practical appeal of popular matchings for the HR and HRLQ problems.
