The planar k-means problem is NP-hard

In the k-means problem, we are given a finite set S of points in ℜm, 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 (2007) [7].