Approximation Algorithms for Clustering with Dynamic Points

TL;DR

This paper introduces approximation algorithms for dynamic clustering problems where points evolve over time and centers can move, providing solutions with provable guarantees and analyzing their computational hardness.

Contribution

It presents the first constant-factor approximation algorithms for dynamic ordered k-median and improved approximations for dynamic k-supplier with outliers, along with hardness results.

Findings

Constant-factor approximation for dynamic ordered k-median.

3-approximation for dynamic k-supplier.

Multi-criteria approximation for outlier version with two time steps.

Abstract

We study two generalizations of classic clustering problems called dynamic ordered $k$-median and dynamic $k$-supplier, where the points that need clustering evolve over time, and we are allowed to move the cluster centers between consecutive time steps. In these dynamic clustering problems, the general goal is to minimize certain combinations of the service cost of points and the movement cost of centers, or to minimize one subject to some constraints on the other. We obtain a constant-factor approximation algorithm for dynamic ordered $k$-median under mild assumptions on the input. We give a 3-approximation for dynamic $k$-supplier and a multi-criteria approximation for its outlier version where some points can be discarded, when the number of time steps is two. We complement the algorithms with almost matching hardness results.