Faster Algorithms for the Constrained k-means Problem

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. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O1, ..., Ok are an arbitrary partition of the dataset and the goal is to output k-centers c1, ..., ck such that the objective function sumi=1k sum_x ∈ Oi ||x - ci||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 varepsilon > 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+varepsilon) 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^tildeO(k/varepsilon) k-centers. We also give a closely matching lower bound of 2^tildeOmega(k/√varepsilon). Moreover, our algorithm runs in time O left(n d · 2^tildeO(k/varepsilon) right). This is a significant improvement over the previous result of Ding and Xu who gave an algorithm with running time O left(n d · (logn)k · 2poly(k/varepsilon) right) and output a list of size O left((logn)k · 2poly(k/varepsilon) right).