How is Time-reversal symmetry important in Computational methods like Density Functional Theory (DFT)? It's well known that symmetries help reduce the computational load in ab-initio calculations. These symmetries could involve rotation, inversion etc. I can understand how these symmetries make the calculation faster but does time reversal symmetry actually improve speed of these calculations? Take the simple case of a ferromagnet.
 A: It depends on what kind of systems we are talking about. Everything below is related to periodic systems. In such systems one has to do a lot of computational work in the reciprocal space, more specifically over the $k$-points in the Brillouin zone (BZ). Imagine a system without any spacial point symmetries (P1). Naively, this implies that one would have to consider all points in the BZ. However, the time reversal symmetry allows us to do integrations/summations/etc. over only a half of the BZ, resulting in the respective reduction in the computation effort.
This was a general consideration. In case of a ferromagnet, one has to be more careful because, strictly speaking, the magnetic moment breaks the time reversal symmetry. But if we neglect spin-orbit coupling (which is often done) then the spin degrees of freedom completely decouple from the spacial degrees of freedom, and we can treat spin-up and spin-down states completely separately. In particular, this implies that you again can use at most half of the BZ. Other spacial symmetries, of course, reduce the effort even more.
If spin-orbit coupling is taken into account, then things get tricky. If the system is magnetic, then one can forget about the time reversal symmetry. So, one would potentially have to work with a larger set of $k$-points in the BZ. If our system is non-magnetic there is still the Kramer's theorem that tells us that spin-up and spin-down states with opposite momenta (i.e., at points $\mathbf{k}$ and $-\mathbf{k}$, respectively) are degenerate in energy. But these states are not related by a simple transformation, and there is no easy way to use this degeneracy to reduce the computational effort in the context of DFT.
