  This paper studies the interference channel with a cognitive relay (ICCR)
under delayed feedback. Three types of delayed feedback are studied: delayed
channel state information at the transmitter (CSIT), delayed output feedback,
and delayed Shannon feedback. Outer bounds are derived for the DoF region of
the two-user multiple-input multiple-output (MIMO) ICCR with delayed feedback
as well as without feedback. For the single-input single-output (SISO)
scenario, optimal schemes are proposed based on retrospective interference
alignment. It is shown that while a cognitive relay without feedback cannot
extend the sum-DoF beyond $1$ in the two-user SISO interference channel,
delayed feedback in the same scenario can extend the sum-DoF to $4/3$. For the
MIMO case, achievable schemes are obtained via extensions of retrospective
interference alignment, leading to DoF regions that meet the respective upper
bounds.
