Local Search for K-medians and Facility Location

Keywords and Synonyms

k-Medians ; k-Means; k-Medioids; Facility location; Point location; Warehouse location; Clustering          

Problem Definition

Clustering is a form of unsupervised learning, where the goal is to “learn” useful patterns in a data set ${ \mathcal{D} }$ of size n. It can also be thought of as a data compression scheme where a large data set is represented using a smaller collection of “representatives”. Such a scheme is characterized by specifying the following:

• 1. A distance metric ${ \mathbf{d} }$ between items in the data set. This metric should satisfy the triangle inequality: ${ \mathbf{d}(i,j) \le \mathbf{d}(j,k) + \mathbf{d}(k,i) }$ for any three items ${ i,j,k \in \mathcal{D} }$. In addition, ${ \mathbf{d}(i,j) = \mathbf{d}(j,i) }$ for all ${ i,j \in \mathcal{S} }$ and ${ \mathbf{d}(i,i) = 0 }$. Intuitively, if the distance between two items is smaller, they are more similar. The items are usually points in some high dimensional Euclidean...