  In recent years, the mathematical and algorithmic aspects of the phase
retrieval problem have received considerable attention. Many papers in this
area mention crystallography as a principal application. In crystallography,
the signal to be recovered is periodic and comprised of atomic distributions
arranged homogeneously in the unit cell of the crystal. The crystallographic
problem is both the leading application and one of the hardest forms of phase
retrieval. We have constructed a graded set of benchmark problems for
evaluating algorithms that perform this type of phase retrieval. The data,
publicly available online, is provided in an easily interpretable format. We
also propose a simple and unambiguous success/failure criterion based on the
actual needs in crystallography. Baseline runtimes were obtained with an
iterative algorithm that is similar but more transparent than those used in
crystallography. Empirically, the runtimes grow exponentially with respect to a
new hardness parameter: the sparsity of the signal autocorrelation. We also
review the algorithms used by the leading software packages. This set of
benchmark problems, we hope, will encourage the development of new algorithms
for the phase retrieval problem in general, and crystallography in particular.
