k-Center Problems with Minimum Coverage

The k-center problem is a well-known facility location problem and can be described as follows: Given a complete undirected graph G=(V,E), a metric d:V×V→ℝ₊ and a positive integer k, we seek a subset U ⊆ V of at most k centers which minimizes the maximum distances from points in V to U. Formally, the objective function is given by: $$ \textstyle\min_{U\subseteq V,|U|\leq k}\max_{v\in V}\min_{r\in U}d(v,r).$$ As a typical example, we may want to set up k service centers (e.g., police stations, fire stations, hospitals, polling centers) and minimize the maximum distances between each client and these centers. The problem is known to be $\mathcal{NP}$-hard [2].