## Abstract

Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric k-Clustering, ℓ_{2}^{2} k-Clustering, and ℓ_{2}^{2} k-Median. In the k-Clustering problem, the objective is to minimize the sum of all intra-cluster distances. In the k-Median problem, the goal is to minimize the sum of distances from points in a cluster to the (best choice of) cluster center. In metric instances, the input distance function is a metric. In ℓ_{2}^{2} instances, the points are in ℝ^{d} and the distance between two points x, y is measured by ||x - y||_{2}^{2} (notice that (ℝ^{d}, || · ||_{2}^{2}) is not a metric space). For the first two problems, our results are the first polynomial time approximation schemes. For the third problem, the running time of our algorithms is a vast improvement over previous work.

Original language | American English |
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Pages (from-to) | 50-58 |

Number of pages | 9 |

Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

State | Published - 2003 |

Externally published | Yes |

Event | 35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: 9 Jun 2003 → 11 Jun 2003 |

## Keywords

- Algorithms
- Theory