  The present paper constructs coresets for weight-constrained anisotropic
assignment and clustering. In contrast to the well-studied unconstrained
least-squares clustering problem, approximating the centroids of the clusters
no longer suffices in the weight-constrained anisotropic case, as even the
assignment of the points to best sites is involved. This assignment step is
often the limiting factor in materials science, a problem that partially
motivates our work. We build on a paper by Har-Peled and Kushal, who
constructed coresets of size
$\mathcal{O}\bigl(\frac{k^3}{\epsilon^{d+1}}\bigr)$ for unconstrained
least-squares clustering. We generalize and improve on their results in various
ways, leading to even smaller coresets with a size of only
$\mathcal{O}\bigl(\frac{k^2}{\epsilon^{d+1}}\bigr)$ for weight-constrained
anisotropic clustering. Moreover, we answer an open question on coreset designs
in the negative, by showing that the total sensitivity can become as large as
the cardinality of the original data set in the constrained case. Consequently,
many techniques based on importance sampling do not apply to weight-constrained
clustering.
