Approximation Algorithms for Spherical k-Means Problem with Penalties Using Local Search Techniques

In this paper, we consider the spherical k-means problem with penalties, a robust model of spherical clusterings that requires identifying outliers during clustering to improve the quality of the solution. Each outlier will incur a specified penalty cost. In this problem, one should detect the outliers and propose a k-clustering for the given data set so as to minimize the sum of the clustering and penalty costs. As our main contribution, we present a (16 + 8 root 3)-approximation via single-swap local search and an (8 + 2 root 7 + ??)-approximation via multi-swap local search.