On robust maximum flow with polyhedral uncertainty sets

We address the problem of determining a robust maximum flow value in a network with uncertain link capacities taken in a polyhedral uncertainty set. Besides a few polynomial cases, we focus on the case where the uncertainty set is taken to be the solution set of an associated (continuous) knapsack problem. This class of problems is shown to be polynomially solvable for planar graphs, but NP-hard for graphs without special structure. The latter result provides evidence of the fact that the problem investigated here has a structure fundamentally different from the robust network flow models proposed in various other published works.