Improved and Simplified Inapproximability for k-means

The k-means problem consists of finding k centers in R that minimize the sum of the squared distances of all points in an input set P from R to their closest respective center. Awasthi et. al. recently showed that there exists a constant ε′ > 0 such that it is NP-hard to approximate the k-means objective within a factor of 1 + ε′. We establish that the constant ε′ is at least 0.0013. For a given set of points P ⊂ R, the k-means problem consists of finding a partition of P into k clusters (C1, . . . , Ck) with corresponding centers (c1, . . . , ck) that minimize the sum of the squared distances of all points in P to their corresponding center, i.e. the quantity arg min (C1,...,Ck),(c1,...,ck) k ∑