90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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In graphical representations of public transportation networks, there is often some degree of uncertainty in the arc values, due to delays or transfer times. This uncertainty can be expressed as a parameterized weight on the transfer arcs. Classical shortest path algorithms often have difficulty handling parameterized arc weights and a tropical geometry approach has been shown as a possible solution. The connection between the classical shortest path problem and tropical geometry is well establish: Tropically multiplying the n × n adjacency matrix of a graph with itself n − 1 times results in the so-called Kleene star, and is a matrix-form solution to the all-pairs shortest path problem. Michael Joswig and Benjamin Schröter showed in their paper The Tropical Geometry of Shortest Paths that the same method can be used to find the solution to the all-pairs shortest path problem even in the case of variable arc weights and they proposed an algorithm to solve the single-target shortest path problem in such a case. The solution takes the form of a polyhedral subdivision of the parameter space. As the number of variable arc weights grows, the time needed to execute an implementation of this algorithm grows exponentially. As the size of a public transportation network grows, the number of variable arc weights grows exponentially as well. However, it has been observed that in public transportation networks, there are usually only a few possible shortest routes. Geometrically, this means that there should be few polyhedra in the polyhedral subdivision. This algorithm is used on an example of a real-world public transportation network and an analysis of the polyhedral subdivision is made. Then a geometrical approach is used to analyze the impact of limiting the number of transfers, and thereby limiting the number of parameterized arcs used, as an estimation of the solution to the all-pairs shortest path problem
We investigate polyhedral aspects of the Periodic Event Scheduling Problem (PESP), the mathematical basis for periodic timetabling problems in public transport. Flipping the orientation of arcs, we obtain a new class of valid inequalities, the flip inequalities, comprising both the known cycle and change-cycle inequalities. For a point of the LP relaxation, a violated flip inequality can be found in pseudo-polynomial time, and even in linear time for a spanning tree solution. Our main result is that the integer vertices of the polytope described by the flip inequalities are exactly the vertices of the PESP polytope, i.e., the convex hull of all feasible periodic slacks with corresponding modulo parameters. Moreover, we show that this flip polytope equals the PESP polytope in some special cases. On the computational side, we devise several heuristic approaches concerning the separation of cutting planes from flip inequalities. These produce better dual bounds for the smallest and largest instance of the benchmarking library PESPlib.
One of the fundamental steps in the optimization of public transport is line planning. It involves determining lines and assigning frequencies of service such that costs are minimized while also maximizing passenger comfort and satisfying travel demands. We formulate the problem as a mixed integer linear program that considers all circuit-like lines in a graph and allows free passenger routing. Traveler and operator costs are included in a linear scalarization in the objective. We apply said programming problem to the Parametric City, which is a graph model introduced by Fielbaum, Jara-Díaz and Gschwender that exibly represents different cities. In his dissertation, Fielbaum solved the line planning problem for various parameter choices in the Parametric City. In a first step, we therefore review his results and make comparative computations. Unlike Fielbaum we arrive at the conclusion that the optimal line plan for this model indeed depends on the demand. Consequently, we analyze the line planning problem in-depth: We find equivalent, but easier to compute formulations and provide a lower bound by LP-relaxation, which we show to be equivalent to a multi-commodity flow problem. Further, we examine what impact symmetry has on the solutions. Supported both by computational results as well as by theoretical analysis, we reach the conclusion that symmetric line plans are optimal or near-optimal in the Parametric City. Restricting the model to symmetric line plans allows for a \kappa-factor approximation algorithm for the line planning problem in the Parametric City.
While graph covering is a fundamental and well-studied problem, this field lacks a broad and unified literature review. The holistic overview of graph covering given in this article attempts to close this gap. The focus lies on a characterization and classification of the different problems discussed in the literature. In addition, notable results and common approaches are also included. Whenever appropriate, our review extends to the corresponding partioning problems.