Improved and Simplified Inapproximability for k-means

TL;DR

This paper improves the understanding of the computational difficulty of approximating the k-means clustering problem by establishing a tighter lower bound on the approximation factor, showing it cannot be approximated within a factor less than 1.0013 unless P=NP.

Contribution

The paper provides a simplified proof establishing that approximating k-means within a factor less than 1.0013 is NP-hard, refining previous bounds.

Findings

Approximation factor c > 1.0013 is NP-hard to achieve.

Simplified proof technique for inapproximability.

Improved bounds on k-means approximation hardness.

Abstract

The k-means problem consists of finding k centers in the d-dimensional Euclidean space that minimize the sum of the squared distances of all points in an input set P to their closest respective center. Awasthi et. al. recently showed that there exists a constant c > 1 such that it is NP-hard to approximate the k-means objective within a factor of c. We establish that the constant c is at least 1.0013.