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

2024-10-15

AI Summary

This paper studies the $k$-center problem in fully dynamic metric spaces: the underlying point set evolves via insertions and deletions, and one must maintain a center set $S$ of size at most $k$ to minimize the maximum covering radius $max_{xin V}min_{yin S}d(x,y)$. We propose the first algorithm achieving simultaneously an $O(1)$-approximation ratio, $O(1)$ amortized center replacements (i.e., updates to $S$), and $ ilde{O}(k)$ amortized update time. Our method integrates Bateni et al.’s dynamic $k$-center framework with Bhattacharya et al.’s dynamic sparsifier construction, overcoming prior trade-offs among approximation quality, stability, and efficiency. All three performance metrics are asymptotically optimal up to logarithmic factors, establishing the first balanced, efficient solution for dynamic clustering under arbitrary point updates.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we consider the emph{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 emph{approximation ratio}, emph{recourse} (the number of changes it makes to the solution $S$), and emph{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 $ ilde O(k)$ amortized update time, emph{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].

Problem

• Maintain dynamic k-center clustering with minimal changes
• Achieve near-optimal approximation ratio and update time
• Handle point insertions and deletions in metric spaces

Innovation

• Dynamic k-center with O(1) approximation
• O(1) amortized recourse for stability
• Near-optimal update time ~O(k)