## Abstract

In many classic clustering problems, we seek to sketch a massive data set of n points (a.k.a clients) in a metric space, by segmenting them into k categories or clusters, each cluster represented concisely by a single point in the metric space (a.k.a. the cluster’s center or its facility). The goal is to find such a sketch that minimizes some objective that depends on the distances between the clients and their respective facilities (the objective is a.k.a. the service cost). Two notable examples are the k-center/k-supplier problem where the objective is to minimize the maximum distance from any client to its facility, and the k-median problem where the objective is to minimize the sum over all clients of the distance from the client to its facility. In practical applications of clustering, the data set may evolve over time, reflecting an evolution of the underlying clustering model. Thus, in such applications, a good clustering must simultaneously represent the temporal data set well, but also not change too drastically between time steps. In this paper, we initiate the study of a dynamic version of clustering problems that aims to capture these considerations. In this version there are T time steps, and in each time step t ∈ {1, 2, . . ., T }, the set of clients needed to be clustered may change, and we can move the k facilities between time steps. The general goal is to minimize certain combinations of the service cost and the facility movement cost, or minimize one subject to some constraints on the other. More specifically, we study two concrete problems in this framework: the Dynamic Ordered k-Median and the Dynamic k-Supplier problem. Our technical contributions are as follows: We consider the Dynamic Ordered k-Median problem, where the objective is to minimize the weighted sum of ordered distances over all time steps, plus the total cost of moving the facilities between time steps. We present one constant-factor approximation algorithm for T = 2 and another approximation algorithm for fixed T ≥ 3. We consider the Dynamic k-Supplier problem, where the objective is to minimize the maximum distance from any client to its facility, subject to the constraint that between time steps the maximum distance moved by any facility is no more than a given threshold. When the number of time steps T is 2, we present a simple constant factor approximation algorithm and a bi-criteria constant factor approximation algorithm for the outlier version, where some of the clients can be discarded. We also show that it is NP-hard to approximate the problem with any factor for T ≥ 3.

Original language | American English |
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Title of host publication | 28th Annual European Symposium on Algorithms, ESA 2020 |

Editors | Fabrizio Grandoni, Grzegorz Herman, Peter Sanders |

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

ISBN (Electronic) | 9783959771627 |

DOIs | |

State | Published - 1 Aug 2020 |

Event | 28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy Duration: 7 Sep 2020 → 9 Sep 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 173 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 28th Annual European Symposium on Algorithms, ESA 2020 |
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Country/Territory | Italy |

City | Virtual, Pisa |

Period | 7/09/20 → 9/09/20 |

### Bibliographical note

Funding Information:Funding Shichuan Deng and Jian Li: Supported in part by the National Natural Science Foundation of China Grant 61822203, 61772297, 61632016, 61761146003, the Zhongguancun Haihua Institute for Frontier Information Technology, Turing AI Institute of Nanjing, and Xi’an Institute for Interdisciplinary Information Core Technology. Yuval Rabani: Supported in part by ISF grant number 2553-17.

Funding Information:

Shichuan Deng and Jian Li: Supported in part by the National Natural Science Foundation of China Grant 61822203, 61772297, 61632016, 61761146003, the Zhongguancun Haihua Institute for Frontier Information Technology, Turing AI Institute of Nanjing, and Xi?an Institute for Interdisciplinary Information Core Technology. Yuval Rabani: Supported in part by ISF grant number 2553-17.

Publisher Copyright:

© Shichuan Deng, Jian Li, and Yuval Rabani

## Keywords

- Clustering
- Dynamic points
- Multi-objective optimization