General $k$-center problem statement: Let \((X, d)\) be a metric space where \(X\) is a set and \(d\) is a metric. A set \(V \subseteq X\) is provided together with a parameter \(k\). The goal is to find a subset \(C \subseteq V\) with \(|C| = k\) such that the maximum distance of a point in \(V\) to the closest point in \(C\) is minimized. The problem can be formally defined as follows:

• Input: a set $V \subseteq X$, and a parameter $k$.
• Output: a set $C \subseteq V$ of $k$ points.
• Goal: Minimize the cost $r^C(V) = \max_{v \in V} d(v, C)$

The $k$-Center Clustering problem: Given a complete undirected graph \(G = (V, E)\) with distances \(d(v_i, v_j) \in \mathbb{N}\) satisfying the triangle inequality, find a subset \(C \subseteq V\) with \(|C| = k\) while minimizing:

$$ \max_{v \in V} \min_{c \in C} d(v, c) $$