On Clustering with Discounts

TL;DR

This paper introduces a new approximation algorithm for the k-median with discounts problem, unifying several clustering problems and improving previous guarantees through iterative LP rounding.

Contribution

It presents the first bi-criteria constant-factor approximation algorithms for k-median with discounts, including matroid and knapsack variants, advancing the theoretical understanding.

Findings

Achieved a bi-criteria constant-factor approximation for k-median with discounts.

Improved the approximation guarantee over previous work.

Extended algorithms to matroid and knapsack median clustering with discounts.

Abstract

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.