Approximation algorithm for spherical $ k $-means problem with penalty

<p style='text-indent:20px;'>The <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-means problem is a classical combinatorial optimization which has lots of applications in many fields such as machine learning, data mining, etc. We consider variant id="M3">\begin{document}$ the spherical space, that is, id="M4">\begin{document}$ with penalties. In problem, it allowable some nodes space can not be clustered by paying penalty costs. Based on local search scheme, we propose id="M5">\begin{document}$ \left(4 (11+4\sqrt{7})+ \epsilon\right) $\end{document}</tex-math></inline-formula>-approximation algorithm using singe-swap operation, where id="M6">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> positive constant.</p>