A Global Optimization Algorithm for K-Center Clustering of One Billion Samples

TL;DR

This paper introduces a scalable global optimization algorithm for K-center clustering that guarantees finding the optimal solution for extremely large datasets within practical time frames, outperforming heuristic methods.

Contribution

A novel reduced-space branch and bound algorithm with a two-stage decomposable lower bound and acceleration techniques for large-scale K-center clustering.

Findings

Solves K-center problems with up to one billion samples within hours.

Achieves 25.8% average reduction in objective function compared to heuristics.

Demonstrates effectiveness on synthetic and real-world datasets.

Abstract

This paper presents a practical global optimization algorithm for the K-center clustering problem, which aims to select K samples as the cluster centers to minimize the maximum within-cluster distance. This algorithm is based on a reduced-space branch and bound scheme and guarantees convergence to the global optimum in a finite number of steps by only branching on the regions of centers. To improve efficiency, we have designed a two-stage decomposable lower bound, the solution of which can be derived in a closed form. In addition, we also propose several acceleration techniques to narrow down the region of centers, including bounds tightening, sample reduction, and parallelization. Extensive studies on synthetic and real-world datasets have demonstrated that our algorithm can solve the K-center problems to global optimal within 4 hours for ten million samples in the serial mode and one billion samples in the parallel mode. Moreover, compared with the state-of-the-art heuristic methods, the global optimum obtained by our algorithm can averagely reduce the objective function by 25.8% on all the synthetic and real-world datasets.