Title: Removing Aspect Ratio on the Running Time for Constrained k-center Clustering
Primary area: optimization
Keywords: Constrained Clustering; Approximation Algorithm

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 Δ (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 Δ, 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 runtime 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 runtime in the data size. Empirical experiments show that our proposed methods achieve better performances compared with the state-of-the-art algorithms on both small and large-scale clustering datasets.

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