Uniform capacitated facility location with outliers/penalties

In this paper, we present a framework to design approximation algorithms for capacitated facility location problems with penalties/outliers. We apply our framework to obtain first approximations for capacitated $k$-facility location problem with penalties (C$k$FLwP) and capacitated facility location problem with outliers (CFLwO), for hard uniform capacities. Our solutions incur slight violations in capacities, ($1+ϵ$) for the problems without cardinality($k$) constraint and ($2+ϵ$) for the problems with the cardinality constraint. For the outlier variant, we also incur a small loss ($1+ϵ$) in outliers. To the best of our knowledge, no results are known for CFLwO and C$k$FLwP in the literature. For uniform facility opening cost, we get rid of violation in capacities for CFLwO. Our approach is based on LP rounding technique.