On Clustering with Discounts
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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.
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