  The notion of critical tuple was introduced by Miklau and Suciu (Gerome
Miklau and Dan Suciu. A formal analysis of information disclosure in data
exchange. J. Comput. Syst. Sci., 73(3):507-534, 2007), who also claimed that
the problem of checking whether a tuple is non-critical is complete for the
second level of the polynomial hierarchy. Kostylev identified an error in the
12-page-long hardness proof. It turns out that the issue is rather fundamental:
the proof can be adapted to show hardness of a relative variant of
tuple-non-criticality, but we have neither been able to prove the original
claim nor found an algorithm for it of lower complexity. In this note we state
formally the relative variant and present an alternative, simplified proof of
its hardness; we also give an NP-hardness proof for the original problem, the
best lower bound we have been able to show. Hence, the precise complexity of
the original critical tuple problem remains open.
