Recoverable Robust Timetables: An Algorithmic Approach on Trees

In the context of scheduling and timetabling, we study a challenging combinatorial problem which is very interesting for both practical and theoretical points of view. The motivation behind it is to cope with scheduled activities which might be subject to unavoidable disruptions, such as delays, occurring during the operational phase. The idea is to preventively plan some extra time for the scheduled activities in order to be "prepared" if a delay occurs, and absorb it without the necessity of rescheduling all the activities from scratch. This realizes the concept of designing robust timetables. During the planning phase, one should also consider recovery features that might be applied at runtime if disruptions occur. This leads to the concept of recoverable robust timetables. In this new concept, it is assumed that recovery capabilities are given as input along with the possible disruptions that must be considered. The main objective is the minimization of the overall needed time. The quality of a robust timetable is measured by the price of robustness, i.e., the ratio between the cost of the robust timetable and that of a nonrobust optimal timetable. We show that finding an optimal solution for this problem is NP-hard even though the topology of the network, which models dependencies among activities, is restricted to trees. However, we manage to design a paeudopolynomial time algorithm based on dynamic programming and apply it on both random networks and real case scenarios provided by Italian railways. We evaluate the effect of robustness on the scheduling of the activities and provide the price of robustness with respect to different scenarios. We experimentally show the practical effectiveness and efficiency of the proposed algorithm.