The bi-criteria seeding algorithms for two variants of k-means problem

The k-means problem is very classic and important in computer science and machine learning, so there are many variants presented depending on different backgrounds, such as the k-means problem with penalties, the spherical k-means clustering, and so on. Since the k-means problem is NP-hard, the research of its approximation algorithm is very hot. In this paper, we apply a bi-criteria seeding algorithm to both k-means problem with penalties and spherical k-means problem, and improve (upon) the performance guarantees given by the k-means++ algorithm for these two problems.