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].

Part of the work by the third author was done when visiting The Institute of Mathematical Sciences, Chennai. He was also supported by NSF CAREER award CCR 0237431.