  We investigate the interplay of band structure topology and localization
properties of Wannier functions. To this end, we extend a recently proposed
compressed sensing based paradigm for the search for maximally localized
Wannier functions [Ozolins et al., PNAS 110, 18368 (2013)]. We develop a
practical toolbox that enables the search for maximally localized Wannier
functions which exactly obey the underlying physical symmetries of a
translationally invariant quantum lattice system under investigation. Most
saliently, this allows us to systematically identify the most localized
representative of a topological equivalence class of band structures, i.e., the
most localized set of Wannier functions that is adiabatically connected to a
generic initial representative. We also elaborate on the compressed sensing
scheme and find a particularly simple and efficient implementation in which
each step of the iteration is an $O(N \log N)$ algorithm in the number of
lattice sites $N$. We present benchmark results on one-dimensional topological
superconductors demonstrating the power of these tools. Furthermore, we employ
our method to address the open question whether compact Wannier functions can
exist for symmetry protected topological states like topological insulators in
two dimensions. The existence of such functions would imply exact flat band
models with strictly finite range hopping. Here, we find numerical evidence for
the absence of such functions. We briefly discuss applications in dissipative
state preparation and in devising variational sets of states for tensor network
methods.
