  In a graph where vertices have preferences over their neighbors, a matching
is called popular if it does not lose a head-to-head election against any other
matching when the vertices vote between the matchings. Popular matchings can be
seen as an intermediate category between stable matchings and maximum-size
matchings. In this paper, we aim to maximize the utility of a matching that is
popular but admits only a few blocking edges.
  For general graphs already finding a popular matching with at most one
blocking edge is NP-complete. For bipartite instances, we study the problem of
finding a maximum-utility popular matching with a bound on the number (or more
generally, the cost) of blocking edges applying a multivariate approach. We
show classical and parameterized hardness results for severely restricted
instances. By contrast, we design an algorithm for instances where preferences
on one side admit a master list, and show that this algorithm is optimal.
