  We study the $k$-median with discounts problem, wherein we are given clients
with non-negative discounts and seek to open at most $k$ facilities. The goal
is to minimize the sum of distances from each client to its nearest open
facility which is discounted by its own discount value, with minimum
contribution being zero. $k$-median with discounts unifies many classic
clustering problems, e.g., $k$-center, $k$-median, $k$-facility $l$-centrum,
etc. We obtain a bi-criteria constant-factor approximation using an iterative
LP rounding algorithm. Our result improves the previously best approximation
guarantee for $k$-median with discounts [Ganesh et al., ICALP'21]. We also
devise bi-criteria constant-factor approximation algorithms for the matroid and
knapsack versions of median clustering with discounts.
