The Planar k-Means Problem is NP-Hard

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In the k-means problem, we are given a finite set S of points in \(\Re^m\), and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [6].