A Local Search Algorithm for Radius-Constrained k-Median

Given a set X of n points in a metric space and a radius R, we consider the combining version of k-center and k-median which is with the same objective as k-median, but with the constraint that any x in X is assigned to a center within the range R. In this paper, we propose a bicriteria approximation algorithm achieving a bicriteria approximation ratio of \((3+\varepsilon , 7)\) , by incorporating local search with the technology of finding key balls with consequent center exchanges therein. Compared to the state-of-the-art approximation ratio of (8, 4) that is based on an LP formulation for k-median, we improve the ratio of the total distance from 8 to \(3+\varepsilon \) at the price of compromising the ratio of radius from 4 to 7.