# Temporal Clustering

## TL;DR

This paper introduces the concept of temporal clustering for sequences of unlabeled point sets, proposing optimization problems and algorithms that balance cluster count, spatial cost, and displacement, with theoretical bounds and limitations.

## Contribution

It formulates a new framework for clustering evolving data, generalizing classical clustering objectives to temporal settings and providing algorithms with theoretical guarantees.

## Key findings

- Developed algorithms balancing cluster number, cost, and displacement.
- Established inapproximability results for temporal clustering.
- Generalized classical clustering objectives to temporal data.

## Abstract

We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological surveys and contagious disease surveillance. In this more general setting existing algorithms for classical (i.e.~static) clustering problems are not applicable anymore. We propose a set of optimization problems which we collectively refer to as 'temporal clustering'. The quality of a solution to a temporal clustering instance can be quantified using three parameters: the number of clusters $k$, the spatial clustering cost $r$, and the maximum cluster displacement $\delta$ between consecutive time steps. We consider spatial clustering costs which generalize the well-studied $k$-center, discrete $k$-median, and discrete $k$-means objectives of classical clustering problems. We develop new algorithms that achieve trade-offs between the three objectives $k$, $r$, and $\delta$. Our upper bounds are complemented by inapproximability results.