Fast Heuristics for Continuous Dynamic Shortest Paths and All-or-nothing Assignment

The dynamic shortest path problem lies at the heart of within-day dynamic traffic assignment. In this paper we address the case of urban road networks with several thousands of short links, where the theoretical paradigm of diachronic graphs that underlies discrete models becomes unfeasible to apply, especially for real time instances. To overcome this limit we reject the discretization of time at the formalization level and focus on the original version of the problem, where each arc of the network is associated with a function having continuous domain and range that yields the exit time for any given entrance time. Within the resulting continuous framework, where every variable is represented as a temporal profile, we will present two formulations of the dynamic network loading map with implicit path enumeration for the case of deterministic route choice. The first one addresses the general case of time-varying arc costs in presence of tolls. The temporal perspective is the exit time from the current node, which yields a model based on temporal layers. The second one addresses the particular case where arc costs are proportional to travel times and the FIFO rule is satisfied. The temporal perspective is the arrival time at the destination, which yields a model based on vehicle trajectories. Then, we develop a computational approach based on the approximation of the temporal profiles as piecewise linear functions of time, which benefits from the possibility to define at the algorithmic level time intervals of any length suitable with the modulation of demand flows. This results in a dramatic improvement of calculation times and memory usage compared to existing assignment algorithms, where instead the length of time intervals defines the accuracy of arc travel times. The heuristics proposed to solve the two formulations of the dynamic shortest path problem and of the corresponding network flow propagation model allow to evaluate the dynamic network loading map for large road networks with small computing resources in the case of deterministic route choice. Moreover, these can also be utilized within a Montecarlo