The Hardness of Approximation of Euclidean kmeans
Abstract
The Euclidean $k$means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of $n$ points in Euclidean space $R^d$, and the goal is to choose $k$ centers in $R^d$ so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general $k$ and a $(1+\epsilon)$approximation which runs in time $poly(n) 2^{O(k/\epsilon)}$. At the other extreme, the only known computational complexity result for this problem is NPhardness [ADHP'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in $R^d$ can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all $k,d$. In this paper we provide the first hardness of approximation for the Euclidean $k$means problem. Concretely, we show that there exists a constant $\epsilon > 0$ such that it is NPhard to approximate the $k$means objective to within a factor of $(1+\epsilon)$. We show this via an efficient reduction from the vertex cover problem on trianglefree graphs: given a trianglefree graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to trianglefree graphs. To show this we transform $G$, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph $H$, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.03316
 Bibcode:
 2015arXiv150203316A
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms