We study two generalizations of classic clustering problems called dynamic ordered k-median and dynamic k-supplier, where the points that need clustering evolve over time, and we are allowed to move the cluster centers between consecutive time steps. In these dynamic clustering problems, the general goal is to minimize certain combinations of the service cost of points and the movement cost of centers, or to minimize one subject to some constraints on the other. We obtain a constant-factor approximation algorithm for dynamic ordered k-median under mild assumptions on the input. We give a 3-approximation for dynamic k-supplier and a multi-criteria approximation for its outlier version where some points can be discarded, when the number of time steps is two. We complement the algorithms with almost matching hardness results.
Bibliographical noteFunding Information:
Shichuan Deng and Jian Li were supported 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 was supported by ISF grant number 2553-17 .
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- Approximation algorithms
- Dynamic points
- Facility location
- Multi-objective optimization