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A common technique in the solution of large or complex optimization problems is the use of micro–macro transformations. In this paper, we carry out a theoretical analysis of such transformations for the track allocation problem in railway networks. We prove that the cumulative rounding technique of Schlechte et al. satisfies two of three natural optimality criteria and that this performance cannot be improved. We also show that under extreme circumstances, this technique can perform inconveniently by underestimating the global optimal value.
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 approximates crossing costs by shadow costs on individual arcs, thus reducing the SPPCC to 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.
Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone
generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where
$1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$.
Several combinatorial problems reduce to a linear optimization problem over $K$.
This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a
completely positive representation of $K$.
We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.
The Flight Planning Problem is to find a minimum fuel trajectory between two airports in a 3D airway network under consideration of the wind. We show that this problem is NP-hard, even in its most basic version. We then present a novel A∗ heuristic, whose potential function is derived from an idealized vertical profile over the remaining flight distance. This potential is, under rather general assumptions, both admissible and consistent and it can be computed efficiently. The method outperforms the state-of-the-art heuristic on real-life instances.