The seeding algorithm for k-means problem with penalties

The k-means problem is a classic NP-hard problem in machine learning and computational geometry. And its goal is to separate the given set into k clusters according to the minimal squared distance. The k-means problem with penalties, as one generalization of k-means problem, allows that some point need not be clustered instead of being paid some penalty. In this paper, we study the k-means problem with penalties by using the seeding algorithm. We propose that the accuracy only involves the ratio of the maximal penalty value to the minimal one. When the penalty is uniform, the approximation factor reduces to the same one for the k-means problem. Moreover, our result generalizes the k-means++ for k-means problem to the penalty version. Numerical experiments show that our seeding algorithm is more effective than the one without using seeding.