  It is well known that most of the common clustering objectives are NP-hard to
optimize. In practice, however, clustering is being routinely carried out. One
approach for providing theoretical understanding of this seeming discrepancy is
to come up with notions of clusterability that distinguish realistically
interesting input data from worst-case data sets. The hope is that there will
be clustering algorithms that are provably efficient on such 'clusterable'
instances. In other words, hope that "Clustering is difficult only when it does
not matter" (CDNM thesis, for short).
  We believe that to some extent this may indeed be the case. This paper
provides a survey of recent papers along this line of research and a critical
evaluation their results. Our bottom line conclusion is that that CDNM thesis
is still far from being formally substantiated. We start by discussing which
requirements should be met in order to provide formal support the validity of
the CDNM thesis. In particular, we list some implied requirements for notions
of clusterability. We then examine existing results in view of those
requirements and outline some research challenges and open questions.
