The k-means problem consists of finding k centers in Rd that minimize the sum of the squared distances of all points in an input set P from Rd 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 1+ε′ is at least 1.0013.
Lee, E., Schmidt, M., & Wright, J. (2017). Improved and simplified inapproximability for k-means. Information Processing Letters, 120, 40–43. https://doi.org/10.1016/j.ipl.2016.11.009