  To mitigate the imbalance in the number of assignees in the
Hospitals/Residents problem, Goko et al. [Goko et al., Maximally Satisfying
Lower Quotas in the Hospitals/Residents Problem with Ties, Proc. STACS 2022,
pp. 31:1--31:20] studied the Hospitals/Residents problem with lower quotas
whose goal is to find a stable matching that satisfies lower quotas as much as
possible. In their paper, preference lists are assumed to be complete, that is,
the preference list of each resident (resp., hospital) is assumed to contain
all the hospitals (resp., residents).
  In this paper, we study a more general model where preference lists may be
incomplete. For four natural scenarios, we obtain maximum gaps of the best and
worst solutions, approximability results, and inapproximability results.
