## 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 | American 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

Funding Information:Funding Sandip Banerjee: Research supported in part by Yuval Rabani’s NSFC-ISF grant 2553-17. Rafail Ostrovsky: Supported in part by DARPA under Cooperative Agreement No: HR0011-20-2-0025, NSF Grant CNS-2001096, US-Israel BSF grant 2015782, Google Faculty Award, JP Morgan Faculty Award, IBM Faculty Research Award, Xerox Faculty Research Award, OKAWA Foundation Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of DARPA, the Department of Defense, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright annotation therein. Yuval Rabani: Research supported in part by NSFC-ISF grant 2553-17 and by NSF-BSF grant 2018687.

Publisher Copyright:

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

## Keywords

- Approximation algorithms
- Clustering
- Primal-dual