An exact algorithm for stable instances of the $ k $-means problem with penalties in fixed-dimensional Euclidean space
<p style='text-indent:20px;'>We study stable instances of the <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-means problem with penalties in fixed-dimensional Euclidean space. An instance of the problem is called <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-stable if this instance exists a sole optimal solution and the solution keeps unchanged when distances and penalty costs are scaled by a factor of no more than <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>. Stable instances of clustering problem have been used to explain why certain heuristic algorithms with poor theoretical guarantees perform quite well in practical. For any fixed <inline-formula><tex-math id="M5">\begin{document}$ \epsilon > 0 $\end{document}</tex-math></inline-formula>, we show that when using a common multi-swap local-search algorithm, a <inline-formula><tex-math id="M6">\begin{document}$ (1+\epsilon) $\end{document}</tex-math></inline-formula>-stable instance of the <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula>-means problem with penalties in fixed-dimensional Euclidean space can be solved accurately in polynomial time.</p>
