The electronic relaxation time cannot be predicted from the band-structure alone, because it depends on additional physics. The dominant mechanism for electron relaxation in materials is usually the interaction with phonons (atomic vibrations). In order to compute the relaxation time, therefore, you need to compute the electron-phonon coupling.
In density functional theory-based software, the electron-phonon matrix elements are usually approximated by the Kohn-Sham particle-phonon interactions, and are computed either semi-analytically (from density functional perturbation theory (DFPT)) or fully numerically (from a finite-displacement calculation). In either case, it is often quite computationally intensive to compute the electron-phonon coupling, since you need to sample the Brillouin zone well for both Kohn-Sham states and phonons. The relaxation time you get also depends on both the energy, E, and momentum, k, of the state.
In many practical cases, the relaxation rate is dominated by phonons with small wavevectors (long wavelengths) and low energies, and a reasonable approximation is to restrict the calculation to those phonons. This is the basis of the deformation potential method, and in particular acoustic deformation potential theory, where we further restrict the calculation to the acoustic phonons (lowest energy phonons).
The key concept in deformation potential theory is to note that if the phonon's wavelength is long enough, the corresponding displacement in any particular unit cell is almost exactly the same as that in adjacent unit cells, and we can model it with a perfectly periodic displacement. For the acoustic modes, all the atoms are moving together and the corresponding displacements may be modelled by straining the unit cell. The calculation itself is almost identical to a calculation of the elastic constants of the material, except that it is the valence and conduction band edges which are important, and how they change under the different acoustic phonons (i.e. the strains).
In addition to the deformation potential itself, you also need to know the effective mass of the electrons which, again, is usually approximated by the effective mass of the Kohn-Sham states, computed from the band-curvature around the valence and conduction band edges.
Computing relaxation times from deformation potential theory is an approximation, but it is often fairly accurate and only requires the strain-response of the N-atom unit cell, not all 3N phonon modes, meaning that it is much quicker to compute and scales as a ground state calculation with the number of atoms.
Review paper for electron-phonon calculations in DFT:
F. Giustino, ``Electron-phonon interactions from first principles'',
Rev. Mod. Phys. 89, 015003
Original deformation potential theory papers:
J. Bardeen and W. Shockley, ``Deformation Potentials and Mobilities in Non-Polar Crystals,'' Physical Review, vol. 80, no. 1, pp. 72-80, Oct 1950.
C. Herring and E. Vogt, ``Transport and Deformation-Potential Theory for Many-Valley Semiconductors with Anisotropic Scattering,'' Physical Review, vol. 101, no. 3, pp. 944-961, 1956.