Robust Flows over Time: Models and Complexity Results

TL;DR

This paper investigates the complexity of robust dynamic network flows with uncertain travel times, showing NP-hardness results, analyzing the limitations of temporally repeated flows, and providing bounds on the worst-case optimality gap.

Contribution

It introduces a mathematical model for robust dynamic flows with travel time uncertainty, proves NP-hardness of feasibility verification, and analyzes the performance of temporally repeated flows under uncertainty.

Findings

Verifying feasibility is NP-hard.

Temporally repeated flows are not optimal under robustness.

Optimality gap is at most O(ηk log T) with a lower bound of Ω(log T).

Abstract

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 $\Gamma$ 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 $O(\eta k \log T)$, where $\eta$ and $k$ are newly introduced instance characteristics and provide a matching lower bound instance with optimality gap $\Omega(\log T)$ and $\eta = k = 1$. The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times.