Improved PTAS for the constrained k-means problem

Abstract

The k-means problem has been paid lots of attention in many fields, and each cluster of the k-means problem always satisfies locality property. In this paper, we study the constrained k-means problem, where the clusters do not satisfy locality property and can be an arbitrary partition of the set of points. Ding and Xu presented a unified framework with running time $O(2^{poly (k/\epsilon )} (\log n)^{k+1} nd)$ by applying uniform sampling and simplex lemma techniques such that a collection of size $O(2^{poly (k/\epsilon )} (\log n)^{k+1})$ of candidate sets containing approximate centers is obtained. Then, the collection is enumerated to get the one that can induce a $(1+\epsilon )$-approximation solution for the constrained k-means problem. By applying $D^2$-sampling technique, Bhattacharya, Jaiswal, and Kumar presented an algorithm with running time $O(2^{{\tilde{O}}(k/\epsilon )}nd)$, which is bounded by $O(2^k( \frac{2123ek}{\epsilon ^3})^{64k/\epsilon }knd)$. The algorithm outputs a collection of size $O(2^k( \frac{2123ek}{\epsilon ^3})^{64k/\epsilon })$ of candidate sets containing approximate centers. In this paper, we present an algorithm with running time $O((\frac{1891ek}{\epsilon ^2})^{8k/\epsilon }nd)$ such that a collection of size $O((\frac{1891ek}{\epsilon ^2})^{8k/\epsilon }n)$ of candidate sets containing approximate centers can be obtained.