Fully Dynamic Consistent $k$-Center Clustering

TL;DR

This paper introduces a deterministic fully dynamic algorithm for consistent k-center clustering that maintains a constant factor approximation with worst-case constant recourse per update, improving over trivial bounds.

Contribution

It presents the first fully dynamic, deterministic algorithm for k-center clustering with worst-case constant recourse per update, advancing beyond previous incremental solutions.

Findings

Achieves worst-case constant recourse per update.

Maintains a constant factor approximate solution.

Works against adaptive adversaries.

Abstract

We study the consistent k-center clustering problem. In this problem, the goal is to maintain a constant factor approximate $k$-center solution during a sequence of $n$ point insertions and deletions while minimizing the recourse, i.e., the number of changes made to the set of centers after each point insertion or deletion. Previous works by Lattanzi and Vassilvitskii [ICML '12] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] showed that in the incremental setting, where deletions are not allowed, one can obtain $k \cdot \textrm{polylog}(n) / n$ amortized recourse for both $k$-center and $k$-median, and demonstrated a matching lower bound. However, no algorithm for the fully dynamic setting achieves less than the trivial $O(k)$ changes per update, which can be obtained by simply reclustering the full dataset after every update. In this work, we give the first algorithm for consistent $k$-center clustering for the fully dynamic setting, i.e., when both point insertions and deletions are allowed, and improves upon a trivial $O(k)$ recourse bound. Specifically, our algorithm maintains a constant factor approximate solution while ensuring worst-case constant recourse per update, which is optimal in the fully dynamic setting. Moreover, our algorithm is deterministic and is therefore correct even if an adaptive adversary chooses the insertions and deletions.