Removing Aspect Ratio on the Running Time for Constrained k-center Clustering

Junyu Huang, Qilong Feng, Ziyun Huang, Jinhui Xu, Jianxin Wang

Keywords: Constrained Clustering; Approximation Algorithm

TL;DR: Avoid guessing clustering radii and remove the aspect ratio from the running time for constrained k-center problems

Abstract: In this paper, we consider the constrained $k$-center problems. Existing algorithms for these problems often rely on optimal radius guessing strategy, leading to an overall running time that is dependent on the aspect ratio $\Delta$ (the ratio between the maximum and minimum pairwise distances). This dependency may potentially limit the scalability of the algorithms for handling large-scale datasets. To overcome the aspect ratio dependency issue, we propose a multi-scaling method. Multi-scaling partitions the clustering instance based on relative distances between data points. It then generates a set of candidate radii whose size is independent of $\Delta$, ensuring the existence of at least one radius that can closely approximate the optimal one for any constrained $k$-center instance. This narrows the search space for radius guessing and removes the running time dependency on the aspect ratio. To further improve the efficiency of multi-scaling, we introduce a problem-specific data reduction method that allows multi-scaling to operate on a smaller unweighted instance while preserving theoretical guarantees. These techniques enable us to obtain approximation results for a series of constrained $k$-center problems with near-linear running time in the data size. Empirical experiments show that our proposed methods achieve better performances compared with the SOTA algorithms on both small and large-scale clustering datasets.

Primary Area: optimization

Submission Number: 24617