  As confusion with lensing B-modes begins to limit experiments that search for
primordial B-mode polarization, robust methods for delensing the CMB
polarization sky are becoming increasingly important. We investigate in detail
the possibility of delensing the CMB with the cosmic infrared background (CIB),
emission from dusty star-forming galaxies that is an excellent tracer of the
CMB lensing signal, in order to improve constraints on the tensor-to-scalar
ratio $r$. We find that the maps of the CIB, such as current Planck satellite
maps at 545 GHz, can be used to remove more than half of the lensing B-mode
power. Calculating optimal combinations of different large-scale-structure
tracers for delensing, we find that co-adding CIB data and external
arcminute-resolution CMB lensing reconstruction can lead to significant
additional improvements in delensing performance. We investigate whether
measurement uncertainty in the CIB spectra will degrade the delensing
performance if no model of the CIB spectra is assumed, and instead the CIB
spectra are marginalized over, when constraining $r$. We find that such
uncertainty does not significantly affect B-mode surveys smaller than a few
thousand degrees. Even for larger surveys it causes only a moderate reduction
in CIB delensing performance, especially if the surveys have high (arcminute)
resolution, which allows self-calibration of the delensing procedure. Though
further work on the impact of foreground residuals is required, our overall
conclusions for delensing with current CIB data are optimistic: this delensing
method can tighten constraints on $r$ by a factor up to $\approx2.2$, and by a
factor up to $\approx4$ when combined with external $\approx 3 \mu$K-arcmin
lensing reconstruction, without requiring the modeling of CIB properties. CIB
delensing is thus a promising method for the upcoming generation of CMB
polarization surveys.
