Point-to-point shortest paths on dynamic time-dependent road networks

The computation of point-to-point shortest paths on time-dependent road networks has many practical applications which are interesting from an industrial point of view. Typically, users are interested in the path leading to their destination which has the smallest travel time among all possible paths; it is natural to model the shortest paths problem on a time-dependent graph, where the arc weights are travel times that depend on the time of day at which the arc is traversed. We study both fully combinatorial methods and mathematical formulation based methods. From a combinatorial point of view, if we impose some restrictions on the arc weights, the problem can be solved in polynomial time with the well known Dijkstra's algorithm. However, applying Dijkstra's algorithm on a graph with several millions of vertices and arcs, such as a continental road network, may require several seconds of CPU time. This is not acceptable for real-time industrial applications; therefore, the need for speedup techniques arises. Bidirectional search is a standard technique to speed up computations on static (i.e. non time-dependent) graphs; however, as the arrival time at the destination is unknown, the cost of time-dependent arcs around the target node cannot be evaluated, thus bidirectional search cannot be directly applied on time-dependent networks. We propose an algorithm based on an asymmetric bidirectional search, which allows the extension to the time-dependent case of hierarchical speedup techniques, well known for static graphs. Our method deals efficiently with dynamic scenarios where arcs weights can change, so that we can take into account real-time and forecast traffic information as soon as it becomes available. We achieve average query times for time-dependent shortest paths computations that were previously only possible on dynamic graphs with static arc costs. We discuss the integration of our algorithm with an existing real-world industrial application. For general arc weight functions, the problem is not polynomially solvable; we propose a mathematical programming formulation which is a Mixed-Integer Linear Program (MILP) if the time-dependent arc weights are linear or piecewise linear functions, whereas it is a Mixed-Integer Nonlinear Program (MINLP) if the arc weights are nonlinear functions. We study efficient algorithms for both classes of problems, and test them on benchmark instances taken from the literature, as well as shortest paths instances. We propose new branching strategies within the context of a Branch-and-Bound algorithm for MILPs. Computational experiments show that, by generating good branching decisions, we enumerate on average half the nodes enumerated by traditional strategies. Our approach is also competitive in terms of total computational time. Finally, we present a general-purpose heuristic for MINLPs based on Variable Neighbourhood Search, Local Branching, Sequential Quadratic Programming and Branch-and-Bound. Experiments show the reliability of our heuristic with respect to methods proposed in the literature