  In the Equal Maximum Flow Problem (EMFP), we aim for a maximum flow where we
require the same flow value on all edges in some given subsets of the edge set.
In this paper, we study the closely related Almost Equal Maximum Flow Problems
(AEMFP) where the flow values on edges of one homologous edge set differ at
most by the valuation of a so called deviation function~$\Delta$. We prove that
the integer almost equal maximum flow problem (integer AEMFP) is in general
$\mathcal{NP}$-complete, and show that even the problem of finding a fractional
maximum flow in the case of convex deviation functions is also
$\mathcal{NP}$-complete. This is in contrast to the EMFP, which is polynomial
time solvable in the fractional case. We provide inapproximability results for
the integral AEMFP. For the integer AEMFP we state a polynomial algorithm for
the constant deviation and concave case for a fixed number of homologous sets.
