  In this paper we analyze the predictions of the forward approximation in some
models which exhibit an Anderson (single-) or many-body localized phase. This
approximation, which consists in summing over the amplitudes of only the
shortest paths in the locator expansion, is known to over-estimate the critical
value of the disorder which determines the onset of the localized phase.
Nevertheless, the results provided by the approximation become more and more
accurate as the local coordination (dimensionality) of the graph, defined by
the hopping matrix, is made larger. In this sense, the forward approximation
can be regarded as a mean field theory for the Anderson transition in infinite
dimensions. The sum can be efficiently computed using transfer matrix
techniques, and the results are compared with the most precise exact
diagonalization results available.
  For the Anderson problem, we find a critical value of the disorder which is
$0.9\%$ off the most precise available numerical value already in 5 spatial
dimensions, while for the many-body localized phase of the Heisenberg model
with random fields the critical disorder $h_c=4.0\pm 0.3$ is strikingly close
to the most recent results obtained by exact diagonalization. In both cases we
obtain a critical exponent $\nu=1$. In the Anderson case, the latter does not
show dependence on the dimensionality, as it is common within mean field
approximations.
  We discuss the relevance of the correlations between the shortest paths for
both the single- and many-body problems, and comment on the connections of our
results with the problem of directed polymers in random medium.
