The Planar k-Means Problem is NP-Hard

Authors:
Meena Mahajan;Prajakta Nimbhorkar;Kasturi Varadarajan
Affiliations:
The Institute of Mathematical Sciences, Chennai, India 600 113;The Institute of Mathematical Sciences, Chennai, India 600 113;The University of Iowa, Iowa City, USA IA 52242-1419
Venue:
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
Year:
2009

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Abstract

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].