  In this paper, we present a framework to design approximation algorithms for
capacitated facility location problems with penalties/outliers using
LP-rounding. Primal-dual technique, which has been particularly successful in
dealing with outliers and penalties, has not been very successful in dealing
with capacities. On the other hand, no primal-dual solution has been able to
break the hardness of capacitated facility location problem(CFLP). LP-rounding
techniques had also not been very successful in dealing with capacities until a
recent work by Grover et al. \cite{GroverGKP18}. Their constant factor
approximation violating the capacities by a small factor ($1 + \epsilon$) is
promising while dealing with capacities.Though LP-rounding has not been very
promising while dealing with penalties and outliers, we successfully apply it
to deal with them along with capacities. That is, our results are obtained by
rounding the solution to the natural LP once again exhibiting the power of
LP-rounding technique. Solutions obtained by LP-rounding are easy to integrate
with other LP-based algorithms.In this paper, we apply our framework to obtain
first constant factor approximations for capacitated facility location problem
with outlier (CFLPO) and capacitated $k$-facility location problem with
penalty(C$k$FLPP) for hard uniform capacities using LP-rounding. Our solutions
incur slight violations in capacities, ($1 + \epsilon$) for the problems
without cardinality($k$) constraint and ($2 + \epsilon$) for the problems with
the cardinality constraint. For the outlier variant, we also incur a small loss
($1 + \epsilon$) in outliers. Due to the hardness of the underlying problems,
the violations are inevitable. Thus we achieve the best possible by rounding
the solution of natural LP for these problems. To the best of our knowledge, no
results are known for CFLPO and C$k$FLPP.
