Almost Optimal Fully Dynamic $k$-Center Clustering with Recourse

Sayan Bhattacharya, Martin Costa, Ermiya Farokhnejad, Silvio Lattanzi, Nikos Parotsidis
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:4196-4209, 2025.

Abstract

In this paper, we consider the metric $k$-center problem in the fully dynamic setting, where we are given a metric space $(V,d)$ evolving via a sequence of point insertions and deletions and our task is to maintain a subset $S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize its approximation ratio, recourse (the number of changes it makes to the solution $S$) and update time (the time it takes to handle an update). We give a simple algorithm for dynamic $k$-center that maintains a $O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$ amortized update time, obtaining near-optimal approximation, recourse and update time simultaneously. We obtain our result by combining a variant of the dynamic $k$-center algorithm of Bateni et al. [SODA’23] with the dynamic sparsifier of Bhattacharya et al. [NeurIPS’23].

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-bhattacharya25a, title = {Almost Optimal Fully Dynamic $k$-Center Clustering with Recourse}, author = {Bhattacharya, Sayan and Costa, Martin and Farokhnejad, Ermiya and Lattanzi, Silvio and Parotsidis, Nikos}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {4196--4209}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/bhattacharya25a/bhattacharya25a.pdf}, url = {https://proceedings.mlr.press/v267/bhattacharya25a.html}, abstract = {In this paper, we consider the metric $k$-center problem in the fully dynamic setting, where we are given a metric space $(V,d)$ evolving via a sequence of point insertions and deletions and our task is to maintain a subset $S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize its approximation ratio, recourse (the number of changes it makes to the solution $S$) and update time (the time it takes to handle an update). We give a simple algorithm for dynamic $k$-center that maintains a $O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$ amortized update time, obtaining near-optimal approximation, recourse and update time simultaneously. We obtain our result by combining a variant of the dynamic $k$-center algorithm of Bateni et al. [SODA’23] with the dynamic sparsifier of Bhattacharya et al. [NeurIPS’23].} }
Endnote
%0 Conference Paper %T Almost Optimal Fully Dynamic $k$-Center Clustering with Recourse %A Sayan Bhattacharya %A Martin Costa %A Ermiya Farokhnejad %A Silvio Lattanzi %A Nikos Parotsidis %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-bhattacharya25a %I PMLR %P 4196--4209 %U https://proceedings.mlr.press/v267/bhattacharya25a.html %V 267 %X In this paper, we consider the metric $k$-center problem in the fully dynamic setting, where we are given a metric space $(V,d)$ evolving via a sequence of point insertions and deletions and our task is to maintain a subset $S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize its approximation ratio, recourse (the number of changes it makes to the solution $S$) and update time (the time it takes to handle an update). We give a simple algorithm for dynamic $k$-center that maintains a $O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$ amortized update time, obtaining near-optimal approximation, recourse and update time simultaneously. We obtain our result by combining a variant of the dynamic $k$-center algorithm of Bateni et al. [SODA’23] with the dynamic sparsifier of Bhattacharya et al. [NeurIPS’23].
APA
Bhattacharya, S., Costa, M., Farokhnejad, E., Lattanzi, S. & Parotsidis, N.. (2025). Almost Optimal Fully Dynamic $k$-Center Clustering with Recourse. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:4196-4209 Available from https://proceedings.mlr.press/v267/bhattacharya25a.html.

Related Material