Clustering under a knapsack constraint: Parameterized approximation for the knapsack median problem

The Knapsack Median problem, defined over a set of clients and facilities in a metric space, seeks to open a subset of facilities and connect each client to an opened facility, with the goal of minimizing the sum of client-connection costs while keeping the sum of facility-opening costs within a specified budget. Solving this problem exactly in FPT time, parameterized by the maximum number of opened facilities (denoted by k), is unlikely due to its W[2]-hardness. Thus, we focus on parameterized approximation algorithms for the problem. We give a sampling-based method that reduces the solution search space, which yields a $(3+\epsilon )$-approximation algorithm running in ${\left(k{\epsilon }^{-1}\right)}^{O\left(k\right)}{n}^{O\left(1\right)}$ time in general metric spaces and a $(1+\epsilon )$-approximation algorithm with similar running time in d-dimensional Euclidean space.