Robust Flows over Time: Models and Complexity Results

We study dynamic network flows with uncertain input data under a robust optimization perspective. In the dynamic maximum flow problem, the goal is to maximize the flow reaching the sink within a given time horizon T , while flow requires a certain travel time to traverse an edge. In our setting, we account for uncertain travel times of flow. We investigate maximum flows over time under the assumption that at most Γ travel times may be prolonged simultaneously due to delay. We develop and study a mathematical model for this problem. As the dynamic robust flow problem generalizes the static version, it is NP-hard to compute an optimal flow. However, our dynamic version is considerably more complex than the static version. We show that it is NP-hard to verify feasibility of a given candidate solution. Furthermore, we investigate temporally repeated flows and show that in contrast to the non-robust case (that is, without uncertainties) they no longer provide optimal solutions for the robust problem, but rather yield a worst case optimality gap of at least T . We finally show that the optimality gap is at most O(ηk log T ), where η and k are newly introduced instance characteristics and provide a matching lower bound instance with optimality gap Ω(log T ) and η = k = 1. The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times.