Abstract In this paper, we consider a class of constrained clustering problems of points in R d , where d could be rather high. A common feature of these problems is that their optimal clusterings no longer have the locality property (due to the additional constraints), which is a key property required by many algorithms for their unconstrained counterparts. To overcome the difficulty caused by the loss of locality, we present in this paper a unified framework, called Peeling-and-Enclosing (PnE), to iteratively solve two variants of the constrained clustering problems, constrained k-means clustering (k-CMeans) and constrained k-median clustering (k-CMedian). Our framework generalizes Kumar et al.'s elegant k-means clustering approach [51] from unconstrained data to constrained data, and is based on two standalone geometric techniques, called Simplex Lemma and Weaker Simplex Lemma, for k-CMeans and kCMedian, respectively. The simplex lemma (or weaker simplex lemma) enables us to efficiently approximate the mean (or median) point of an unknown set of points by searching a small-size grid, independent of the dimensionality of the space, in a simplex (or the surrounding region of a simplex), and thus can This work was supported in part by NSF through grants IIS-1115220, IIS-1422591, CCF-1422324, CNS-1547167, CCF-1656905, and CCF-1716400. The first author was also supported by a start-up fund from Michigan State University. A preliminary version of this paper has appeared in Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015) 1 are fixed numbers, our framework generates, in nearly linear time (i.e., O(n(log n) k+1 d)), O((log n) k ) k-tuple candidates for the k mean or median points, and one of them induces a (1 + )-approximation for k-CMeans or k-CMedian, where n is the number of points. Combining this unified framework with a problem-specific selection algorithm (which determines the best k-tuple candidate), we obtain a (1 + )-approximation for each of the constrained clustering problems. Our framework improves considerably the best known results for these problems. We expect that our technique will be applicable to other constrained clustering problems without locality.
Ding and Xu REF provided a unified framework for solving a class of constrained k-median and k-means clusterings.
A Unified Framework for Clustering Constrained Data without Locality Property
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Mathematics Computer Science