Fully Dynamic k-Clustering with Fast Update Time and Small Recourse
Abstract
In the dynamic metric $k$-median problem, we wish to maintain a set of $k$ centers $S \subseteq V$ in an input metric space $(V, d)$ that gets updated via point insertions/deletions, so as to minimize the objective $\sum_{x \in V} \min_{y \in S} d(x, y)$. The quality of a dynamic algorithm is measured in terms of its approximation ratio, ``recourse'' (the number of changes in $S$ per update) and ``update time'' (the time it takes to handle an update). The ultimate goal in this line of research is to obtain a dynamic $O(1)$ approximation algorithm with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time.
Dynamic $k$-median is a canonical example of a class of problems known as dynamic $k$-clustering, that has received significant attention in recent years [Fichtenberger et al, SODA'21], [Bateni et al, SODA'23], [Lacki et al, SODA'24]. To the best of our knowledge, however, all these previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic $k$-median in particular, the state-of-the-art results get $\tilde{O}(k^2)$ update time and $O(k)$ recourse~[Cohen-Addad et al, ICML'19], [Henzinger and Kale, ESA'20], [Bhattacharya et al, NeurIPS'23]. But, this recourse bound of $O(k)$ can be trivially obtained by recomputing an optimal solution from scratch after every update, provided we ignore the update time. In addition, the update time of $\tilde{O}(k^2)$ is polynomially far away from the desired bound of $\tilde{O}(k)$. We come {\em arbitrarily close} to resolving the main open question on this topic, with the following results.
(I) We develop a new framework of {\em randomized local search} that is suitable for adaptation in a dynamic setting. For every $\epsilon > 0$, this gives us a dynamic $k$-median algorithm with $O(1/\epsilon)$ approximation ratio, $\tilde{O}(k^{\epsilon})$ recourse and $\tilde{O}(k^{1+\epsilon})$ update time. This framework also generalizes to dynamic $k$-clustering with $\ell^p$-norm objectives. As a corollary, we obtain similar bounds for the dynamic $k$-means problem, and a new trade-off between approximation ratio, recourse and update time for the dynamic $k$-center problem.
(II) If it suffices to maintain only an estimate of the {\em value} of the optimal $k$-median objective, then we obtain a $O(1)$ approximation algorithm with $\tilde{O}(k)$ update time. We achieve this result via adapting the Lagrangian Relaxation framework of [Jain and Vazirani, JACM'01], and a facility location algorithm of [Mettu and Plaxton, FOCS'00] in the dynamic setting.
Dynamic $k$-median is a canonical example of a class of problems known as dynamic $k$-clustering, that has received significant attention in recent years [Fichtenberger et al, SODA'21], [Bateni et al, SODA'23], [Lacki et al, SODA'24]. To the best of our knowledge, however, all these previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic $k$-median in particular, the state-of-the-art results get $\tilde{O}(k^2)$ update time and $O(k)$ recourse~[Cohen-Addad et al, ICML'19], [Henzinger and Kale, ESA'20], [Bhattacharya et al, NeurIPS'23]. But, this recourse bound of $O(k)$ can be trivially obtained by recomputing an optimal solution from scratch after every update, provided we ignore the update time. In addition, the update time of $\tilde{O}(k^2)$ is polynomially far away from the desired bound of $\tilde{O}(k)$. We come {\em arbitrarily close} to resolving the main open question on this topic, with the following results.
(I) We develop a new framework of {\em randomized local search} that is suitable for adaptation in a dynamic setting. For every $\epsilon > 0$, this gives us a dynamic $k$-median algorithm with $O(1/\epsilon)$ approximation ratio, $\tilde{O}(k^{\epsilon})$ recourse and $\tilde{O}(k^{1+\epsilon})$ update time. This framework also generalizes to dynamic $k$-clustering with $\ell^p$-norm objectives. As a corollary, we obtain similar bounds for the dynamic $k$-means problem, and a new trade-off between approximation ratio, recourse and update time for the dynamic $k$-center problem.
(II) If it suffices to maintain only an estimate of the {\em value} of the optimal $k$-median objective, then we obtain a $O(1)$ approximation algorithm with $\tilde{O}(k)$ update time. We achieve this result via adapting the Lagrangian Relaxation framework of [Jain and Vazirani, JACM'01], and a facility location algorithm of [Mettu and Plaxton, FOCS'00] in the dynamic setting.
