On the Impact of Uncertainty on some Optimization Problems: Combinatorial Aspects of Delay Management and Robust Online Scheduling

Real-life optimization problems frequently rely on input data which only estimates the data actually appearing when a computed solution is used in practice. Small deviations of the actual data from the input data can have many different consequences: for instance, a computed solution might need to be adapted in order to guarantee feasibility, or the quality of a computed solution might degrade significantly. In some cases, a computed solution can be adapted to incorporate the deviations, thus making the solution feasible again or qualitatively better. In other cases, it might be too late to perform adaptations, and the quality of the solution irredeemably suffers from the deviations. In this thesis, we consider two problems dealing with such perturbations of the input data. The first problem considered is delay management in railway systems, a problem arising when trains run late. Delay management consists in deciding which trains should wait for delayed transferring passengers, with the goal of minimizing the passenger discomfort. If a connecting train waits for delayed passengers, these passengers are able to board the connecting train, but delay propagates to all the passengers using the connecting train. As a consequence, these passengers might also miss a future connection. If the connecting train departs as scheduled, the delayed transferring passengers miss the connection, and have to board the next available connection, which results in a bigger delay for them. Delay management aims at finding a trade-off between delaying passengers which were originally on time, and letting delayed passengers miss their connections. Currently, this aspect of railway operations is usually handled by human dispatchers. This thesis analyzes a theoretical model for delay management. This model is highly simplified and not directly applicable in practice. Because of its simplicity, the model allows to focus on the propagation of delays through the network. For this focus and the offline case where the complete input is given in advance, we show that variations of the following parameters have a significant impact on the computational complexity of delay management: the maximum number of transfers over all passengers; the structure of the railway network; the possibility of trains to catch up on their delay; the existence of intermediate stops. The proofs of hardness are complemented with polynomial-time algorithms. We also address the online case, where the information about the delays of passengers is