  We consider the problem of computing popular matchings in a bipartite graph G
= (R U H, E) where R and H denote a set of residents and a set of hospitals
respectively. Each hospital h has a positive capacity denoting the number of
residents that can be matched to h. The residents and the hospitals specify
strict preferences over each other. This is the well-studied Hospital Residents
(HR) problem which is a generalization of the Stable Marriage (SM) problem. The
goal is to assign residents to hospitals optimally while respecting the
capacities of the hospitals. Stability is a well-accepted notion of optimality
in such problems. However, motivated by the need for larger cardinality
matchings, alternative notions of optimality like popularity have been
investigated in the SM setting. In this paper, we consider a generalized HR
setting -- namely the Laminar Classified Stable Matchings (LCSM+) problem.
Here, additionally, hospitals can specify classifications over residents in
their preference lists and classes have upper quotas. We show the following new
results: We define a notion of popularity and give a structural
characterization of popular matchings for the LCSM+ problem. Assume n = |R| +
|H| and m = |E|. We give an O(mn) time algorithm for computing a maximum
cardinality popular matching in an LCSM+ instance. We give an O(mn^2) time
algorithm for computing a matching that is popular amongst the maximum
cardinality matchings in an LCSM+ instance.
