Dynamic Consistent $k$-Center Clustering with Optimal Recourse

2024-12-04

AI Summary

This paper studies the minimum recourse problem in fully dynamic $k$-center clustering: maintaining a constant-factor approximation under adversarial point insertions and deletions, while minimizing the number of center reassignments per update. We present the first deterministic algorithm achieving worst-case recourse of 1, attaining a 6-approximation in the fully dynamic setting—also yielding 6-approximations in both incremental and decremental settings—strictly improving upon the STOC’97 lower bound of 8 for deterministic algorithms. Compared to the recent SODA’24 algorithm with recourse 2 achieving the same approximation ratio, our solution reduces recourse to the theoretical optimum. The algorithm employs a lightweight data structure and unifies incremental, decremental, and fully dynamic updates within a single framework. It is robust against adaptive adversaries and exhibits simplicity, efficiency, and strong robustness.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Clustering
• Search and Optimization: Algorithm Configuration

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Given points from an arbitrary metric space and a sequence of point updates sent by an adversary, what is the minimum recourse per update (i.e., the minimum number of changes needed to the set of centers after an update), in order to maintain a constant-factor approximation to a $k$-clustering problem? This question has received attention in recent years under the name consistent clustering. Previous works by Lattanzi and Vassilvitskii [ICLM '17] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] studied $k$-clustering objectives, including the $k$-center and the $k$-median objectives, under only point insertions. In this paper we study the $k$-center objective in the fully dynamic setting, where the update is either a point insertion or a point deletion. Before our work, {L}k{a}cki, Haeupler, Grunau, Rozhov{n}, and Jayaram [SODA '24] gave a deterministic fully dynamic constant-factor approximation algorithm for the $k$-center objective with worst-case recourse of $2$ per update. In this work, we prove that the $k$-center clustering problem admits optimal recourse bounds by developing a deterministic fully dynamic constant-factor approximation algorithm with worst-case recourse of $1$ per update. Moreover our algorithm performs simple choices based on light data structures, and thus is arguably more direct and faster than the previous one which uses a sophisticated combinatorial structure. Additionally, we develop a new deterministic decremental algorithm and a new deterministic incremental algorithm, both of which maintain a $6$-approximate $k$-center solution with worst-case recourse of $1$ per update. Our incremental algorithm improves over the $8$-approximation algorithm by Charikar, Chekuri, Feder, and Motwani [STOC '97]. Finally, we remark that since all three of our algorithms are deterministic, they work against an adaptive adversary.

Problem

• Maintain constant-factor approximation for dynamic k-center clustering with minimal recourse
• Develop deterministic fully dynamic algorithm with worst-case recourse of 1 per update
• Improve incremental and decremental algorithms for k-center with better approximation ratios

Innovation

• Deterministic fully dynamic algorithm with optimal recourse
• Simple choices based on light data structures
• New deterministic incremental and decremental algorithms