We study the Flight Planning Problem for a single aircraft, where we look for a minimum cost path in the airway network, a directed graph. Arc evaluation, such as weather computation, is computationally expensive due to non-linear functions, but required for exactness. We propose several pruning methods to thin out the search space for Dijkstra’s algorithm before the query commences. We do so by using innate problem characteristics such as an aircraft’s tank capacity, lower and upper bounds on the total costs, and in particular, we present a method to reduce the search space even in the presence of regional crossing costs. We test all pruning methods on real-world instances, and show that incorporating crossing costs into the pruning process can reduce the number of nodes by 90% in our setting.

Let be a directed acyclic graph with arcs, a source and a sink . We introduce the cone of flow matrices, which is a polyhedral cone generated by the matrices , where is the incidence vector of the path . We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc‐representation of a flow, reduce to a linear optimization problem over . This cone is intractable: we prove that the membership problem associated to is NP‐complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial‐time the decomposition of a matrix as a linear combination of some 's. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of . We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc‐capacities.

Real world routing problems, e.g., in the airline industry or in public and rail transit, can feature complex non-linear cost functions. An important case are costs for crossing regions, such as countries or fare zones. We introduce the shortest path problem with crossing costs (SPPCC) to address such situations; it generalizes the classical shortest path problem and variants such as the resource constrained shortest path problem and the minimum label path problem. Motivated by an application in flight trajectory optimization with overflight costs, we focus on the case in which the crossing costs of a region depend only on the nodes used to enter or exit it. We propose an exact Two-Layer-Dijkstra Algorithm as well as a novel cost-projection linearization technique that transforms crossing costs into shadow costs on individual arcs, thus approximating the SPPCC by a standard shortest path problem. We evaluate all algorithms' performance on real-world flight trajectory optimization instances, obtaining very good à posteriori error bounds.

We study the Flight Planning Problem for a single aircraft, which deals with finding a path of minimal travel time in an airway network. Flight time along arcs is affected by wind speed and direction, which are functions of time. We consider three variants of the problem, which can be modeled as, respectively, a classical shortest path problem in a metric space, a time-dependent shortest path problem with piecewise linear travel time functions, and a time-dependent shortest path problem with piecewise differentiable travel time functions. The shortest path problem and its time-dependent variant have been extensively studied, in particular, for road networks. Airway networks, however, have different characteristics: the average node degree is higher and shortest paths usually have only few arcs. We propose A* algorithms for each of the problem variants. In particular, for the third problem, we introduce an application-specific "super-optimal wind" potential function that overestimates optimal wind conditions on each arc, and establish a linear error bound. We compare the performance of our methods with the standard Dijkstra algorithm and the Contraction Hierarchies (CHs) algorithm. Our computational results on real world instances show that CHs do not perform as well as on road networks. On the other hand, A* guided by our potentials yields very good results. In particular, for the case of piecewise linear travel time functions, we achieve query times about 15 times shorter than CHs.