Competitively Consistent Clustering

2025-08-14

AI Summary

This paper studies the $k$-center, facility location, and $k$-median clustering problems in fully dynamic environments, where data points arrive and depart in real time, requiring continuous maintenance of approximate optimal solutions while minimizing the total number of center adjustments. The authors innovatively adapt the Positive Body Chasing framework to online clustering, maintaining slightly more than $k$ centers to jointly optimize approximation ratio and adjustment cost. They introduce a deterministic rounding technique based on fractional solutions and integrate offline optimal adjustment cost analysis to achieve an $O(eta)$-approximation with total adjustment cost $O(log |F| cdot log Delta) cdot mathrm{OPT}_{ ext{tec}}^eta$, where $|F|$ is the number of candidate facilities and $Delta$ is the aspect ratio. Moreover, they establish an $Omega(log |F|)$ lower bound on adjustment cost, demonstrating that their result is nearly tight.

Technology Category

• Machine Learning: Clustering
• Search and Optimization: Algorithm Configuration
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In fully-dynamic consistent clustering, we are given a finite metric space $(M,d)$, and a set $Fsubseteq M$ of possible locations for opening centers. Data points arrive and depart, and the goal is to maintain an approximately optimal clustering solution at all times while minimizing the recourse, the total number of additions/deletions of centers over time. Specifically, we study fully dynamic versions of the classical $k$-center, facility location, and $k$-median problems. We design algorithms that, given a parameter $βgeq 1$, maintain an $O(β)$-approximate solution at all times, and whose total recourse is bounded by $O(log |F| log Δ) cdot ext{OPT}_ ext{rec}^β$. Here $ ext{OPT}_ ext{rec}^β$ is the minimal recourse of an offline algorithm that maintains a $β$-approximate solution at all times, and $Δ$ is the metric aspect ratio. Finally, while we compare the performance of our algorithms to an optimal solution that maintains $k$ centers, our algorithms are allowed to use slightly more than $k$ centers. We obtain our results via a reduction to the recently proposed Positive Body Chasing framework of [Bhattacharya, Buchbinder, Levin, Saranurak, FOCS 2023], which we show gives fractional solutions to our clustering problems online. Our contribution is to round these fractional solutions while preserving the approximation and recourse guarantees. We complement our positive results with logarithmic lower bounds which show that our bounds are nearly tight.

Problem

• Maintain approximate clustering solutions with minimal center changes
• Address dynamic k-center, facility location, and k-median problems
• Round fractional solutions while preserving approximation guarantees

Innovation

• Dynamic clustering with bounded recourse
• Reduction to Positive Body Chasing framework
• Rounding fractional solutions with guarantees