  The robust minimum cost flow problem under consistent flow constraints
(RobMCF$\equiv$) is a new extension of the minimum cost flow (MCF) problem. In
the RobMCF$\equiv$ problem, we consider demand and supply that are subject to
uncertainty. For all demand realizations, however, we require that the flow
value on an arc needs to be equal if it is included in the predetermined arc
set given. The objective is to find feasible flows that satisfy the equal flow
requirements while minimizing the maximum occurring cost among all demand
realizations.
  In the case of a discrete set of scenarios, we derive structural results
which point out the differences with the polynomial time solvable MCF problem
on networks with integral capacities. In particular, the Integral Flow Theorem
of Dantzig and Fulkerson does not hold. For this reason, we require integral
flows in the entire paper. We show that the RobMCF$\equiv$ problem is strongly
$\mathcal{NP}$-hard on acyclic digraphs by a reduction from the $(3,B2)$-Sat
problem. Further, we demonstrate that the RobMCF$\equiv$ problem is weakly
$\mathcal{NP}$-hard on series-parallel digraphs by providing a reduction from
Partition and a pseudo-polynomial algorithm based on dynamic programming.
Finally, we propose a special case on series-parallel digraphs for which we can
solve the RobMCF$\equiv$ problem in polynomial time.
