## Abstract

We give a constant factor polynomial time pseudo-approximation algorithm for min-sum clustering with or without outliers. The algorithm is allowed to exclude an arbitrarily small constant fraction of the points. For instance, we show how to compute a solution that clusters 98% of the input data points and pays no more than a constant factor times the optimal solution that clusters 99% of the input data points. More generally, we give the following bicriteria approximation: For any ϵ > 0, for any instance with n input points and for any positive integer n^{′} ≤ n, we compute in polynomial time a clustering of at least (1 − ϵ)n^{′} points of cost at most a constant factor greater than the optimal cost of clustering n^{′} points. The approximation guarantee grows with ^{1}_{ϵ}. Our results apply to instances of points in real space endowed with squared Euclidean distance, as well as to points in a metric space, where the number of clusters, and also the dimension if relevant, is arbitrary (part of the input, not an absolute constant).

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021 |

Editors | Mary Wootters, Laura Sanita |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772075 |

DOIs | |

State | Published - 1 Sep 2021 |

Event | 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 - Virtual, Seattle, United States Duration: 16 Aug 2021 → 18 Aug 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 207 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 |
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Country/Territory | United States |

City | Virtual, Seattle |

Period | 16/08/21 → 18/08/21 |

### Bibliographical note

Publisher Copyright:© Sandip Banerjee, Rafail Ostrovsky, and Yuval Rabani; licensed under Creative Commons License CC-BY 4.0

## Keywords

- Approximation algorithms
- Clustering
- Primal-dual