An Approximation Algorithm Based on Seeding Algorithm for Fuzzy k-Means Problem with Penalties

As a classic NP-hard problem in machine learning and computational geometry, the k-means problem aims to partition the given dataset into k clusters according to the minimal squared Euclidean distance. Different from k-means problem and most of its variants, fuzzy k-means problem belongs to the soft clustering problem, where each given data point has relationship to every center point. Compared to fuzzy k-means problem, fuzzy k-means problem with penalties allows that some data points need not be clustered instead of being paid penalties. In this paper, we propose an O(αklnk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\alpha k\ln ~k)$$\end{document}-approximation algorithm based on seeding algorithm for fuzzy k-means problem with penalties, where α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} involves the ratio of the maximal penalty value to the minimal one. Furthermore, we implement numerical experiments to show the effectiveness of our algorithm.