Ebooks on Combinatorics

Mon, 07 Jul 2025 00:00:00 +0000

Metric $k$-center

Mon, 07 Jul 2025 00:00:00 +0000

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 can also be defined on a complete undirected graph $G = (V, E)$ as follows:

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:

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

Mathematics - Combinatorics on Nam Le

Ebooks on Combinatorics

Metric $k$-center