  This paper studies a variant of the minimum-cost flow problem in a graph with
convex cost function where the demands at the vertices are functions depending
on a one-dimensional parameter $\lambda$. We devise two algorithmic approaches
for the approximate computation of parametric solutions for this problem. The
first approach transforms an instance of the parametric problem into an
instance with piecewise quadratic cost functions by interpolating the marginal
cost functions. The new instance can be solved exactly with an algorithm we
developed in prior work. In the second approach, we compute a fixed number of
non-parametric solutions and interpolate the resulting flows yielding an
approximate solution for the original, parametric problem. For both methods we
formulate explicit bounds on the step sizes used in the respective
interpolations that guarantee relative and absolute error margins. Finally, we
test our approaches on real-world traffic and gas instances in an empirical
study.
