Streaming PTAS for Constrained k-Means

Abstract: We generalise the results of Bhattacharya et al. (Journal of Computing Systems, 62(1):93-115, 2018) for the list-$k$-means problem defined as -- for a (unknown) partition $X_1, ..., X_k$ of the dataset $X \subseteq \mathbb{R}<sup>d$, find a list of $k$-center sets (each element in the list is a set of $k$ centers) such that at least one of $k$-center sets ${c_1, ..., c_k}$ in the list gives an $(1+\varepsilon)$-approximation with respect to the cost function $$\min_{\textrm{permutation } \pi} \left[ \sum_{i=1}^{k} \sum_{x \in X_i} ||x - c_{\pi(i)}||² \right]$$. The list-$k$-means problem is important for the constrained $k$-means problem since algorithms for the former can be converted to PTAS for various versions of the latter. Following are the consequences of our generalisations: - Streaming algorithm: Our $D<sup>2$-sampling based algorithm running in a single iteration allows us to design a 2-pass, logspace streaming algorithm for the list-$k$-means problem. This can be converted to a 4-pass, logspace streaming PTAS for various constrained versions of the $k$-means problem. - Faster PTAS under stability: Our generalisation is also useful in $k$-means clustering scenarios where finding good centers becomes easy once good centers for a few "bad" clusters have been chosen. One such scenario is clustering under stability where the number of such bad clusters is a constant. Using the above idea, we significantly improve the running time of the known algorithm from $$O(dn³⁾ (k \log{n})^{{poly(\frac{1}{\beta},} \frac{1}{\varepsilon})}$$ to $$O \left(dn³ k^{{\tilde{O}_{\beta} \varepsilon}(\frac{1}{\beta \varepsilon})} \right)$$.