Algorithms and Complexity for the Almost Equal Maximum Flow Problem

In the Equal Maximum Flow Problem (EMFP), we aim for a maximum flow where we require the same flow value on all arcs in some given subsets of the arc set. We study the related Almost Equal Maximum Flow Problems (AEMFP) where the flow values on arcs of one homologous arc set differ at most by the valuation of a so called homologous function Δ. We prove that the integer AEMFP is in general \(\mathcal {N}\mathcal {P}\)-complete, and that even finding a fractional maximum flow in the case of convex homologous functions is also \(\mathcal {N}\mathcal {P}\)-complete. This is in contrast to the EMFP, which is polynomial time solvable in the fractional case. We also 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.