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Faster Algorithms for the Constrained k-Means Problem Bhattacharya, Anup Jaiswal, Ragesh Kumar, Amit k-means k-median approximation algorithm sampling The classical center based clustering problems such as k-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. For instance, consider the r-gather clustering problem where there is an additional constraint that each of the clusters should have at least r points or the capacitated clustering problem where there is an upper bound on the cluster sizes. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O_1, ..., O_k are an arbitrary partition of the dataset and the goal is to output k-centers c_1, ..., c_k such that the objective function sum_{i=1}^{k} sum_{x in O_{i}} ||x - c_{i}||^2 is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of k centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such k centers such that at least one of these k centers behaves well. Given an error parameter epsilon > 0, let l denote the size of the smallest list of k-centers such that at least one of the k-centers gives a (1+epsilon) approximation w.r.t. the objective function above. In this paper, we show an upper bound on l by giving a randomized algorithm that outputs a list of 2^{~O(k/epsilon)} k-centers. We also give a closely matching lower bound of 2^{~Omega(k/sqrt{epsilon})}. Moreover, our algorithm runs in time O(n * d * 2^{~O(k/epsilon)}). This is a significant improvement over the previous result of Ding and Xu who gave an algorithm with running time O(n * d * (log{n})^{k} * 2^{poly(k/epsilon)}) and output a list of size O((log{n})^k * 2^{poly(k/epsilon)}). Our techniques generalize for the k-median problem and for many other settings where non-Euclidean distance measures are involved. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Anup Bhattacharya and Ragesh Jaiswal and Amit Kumar 2016 Is Part Of LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2016.16 urn:nbn:de:0030-drops-57179 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.16 eng https://creativecommons.org/licenses/by/3.0/legalcode