In the freight car dispatching problem, empty freight cars have to be assigned to known demands respecting a given time horizon and certain constraints. The goal is to minimize the resulting transportation costs. One of the constraints is that customers can specify the type of cars they want. It is possible, however, that cars of certain types can be substituted by other cars, either in a 1‐to‐1 fashion or at different exchange rates. We show that these substitutions make the dispatching problem hard to solve and hard to approximate. We model the dispatching problem as an integral generalized transportation problem on a bipartite graph. Using rounding techniques, the LP‐relaxation can be transformed to a transportation schedule violating some of the constraints slightly. Under an additional assumption on the cost function, we fix this violation and derive a 4‐approximation of the problem. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(1), 33–39 2015

We consider the empty freight car distribution problem (DP) at DB Schenker Rail Deutschland AG under a wide range of application relevant constraints and real data sets. The (DP) is an online assignment problem between geographically distributed empty freight car supplies and customer demands for such cars in preparation of good transport. The objective is to minimize transport costs for empty cars while distributing them effectively with respect to the constraints. In our case, one major constraint is given by prescheduled freight trains: obviously a supply can only be assigned to a demand if it reaches the latter in time. Further, the variety of goods (bulk cargo, steel coils, etc.) to be transported requires distinct types of freight cars. Freight cars of a certain type can be exchanged by cars of other types with respect to a given substitution scheme and different 'exchange rates'. Allowed substitutions are therefore another major constraint of the (DP). We describe further `hard' and `soft' constraints and sketch the current work flow at DB Schenker Rail Deutschland AG to find an adequate solution for the (DP) on a daily base in practice. The (DP) is currently solved separately for groups of car types and in several steps. Moreover, some steps contain manual pre- and post-processing to ensure certain constraints. Hence global sub-optimal distributions can occur. We therefore integrate all constraints into a generalized network flow model for the (DP). A global optimal distribution is then provided by an integral minimum cost flow in the network. To find such a flow is NP-hard in general. We show that a general substitution scheme makes our notion of the (DP) also NP-hard. Hence independent of the applied model and with respect to practical runtime requirements, we have to find a compromise between solution time and quality. We do so in two ways. Instances of the (DP) which correspond to classical flow networks are solved by an integral minimum cost flow, which can be obtained in polynomial time. We use such instances to polynomially obtain minimum cost flows of fixed bounded fractionality for certain general instances. For those instances occurring in the application we obtain half-integral flows, which can be rounded to approximate or heuristic distributions in linear time. Moreover, we develop a network-based reoptimization approach, which yields optimal solutions for subsequent instances with few changes very fast. This thesis was inspired and funded by a 2-year research and development project of DB Schenker Rail Deutschland AG in cooperation with the work group Faigle/Schrader of the University of Cologne and the work group of Prof. Dr. Sven O. Krumke at the Technical University of Kaiserslautern. The project included the implementation of the generalized network model and the reoptimization, approximation and heuristic methods. The software is designed as a future optimization kernel for the (DP) at DB Schenker Rail Deutschland AG.

We consider the Minimum Manhattan Subnetwork (MMSN) Problem which generalizes the already known Minimum Manhattan Network (MMN) Problem: Given a set P of n points in the plane, find shortest rectilinear paths between all pairs of points. These paths form a network, the total length of which has to be minimized. From a graph theoretical point of view, a MMN is a 1-spanner with respect to the L1 metric. In contrast to the MMN problem, a solution to the MMSN problem does not demand L1-shortest paths for all point pairs, but only for a given set R⊆P×P of pairs. The complexity status of the MMN problem is still unsolved in ≥2 dimensions, whereas the MMSN was shown to be NP-complete considering general relations R in the plane. We restrict the MMSN problem to transitive relations RT (Transitive Minimum Manhattan Subnetwork (TMMSN) Problem) and show that the TMMSN problem in 3 dimensions is NP-complete.

We introduce a type of movement constraints for a swarm of robots in a grid environment, which is inspired by Alex Randolph’s board game Ricochet Robot and new to the field of robot motion planning. This type of movement may be used to model robots with very limited abilities for self localization: We assume that once a robot starts to drive in a certain direction, it does not stop its movement until it hits an obstacle wall or another robot. We give some lower bounds on the number of robots needed to reach every cell. Especially, it is easy to see that three robots are necessary and sufficient to reach every cell in a simple rectangular environment. Further, we consider the question, whether a certain cell can be reached is in arbitrary environments. A Java applet for simulating robot swarms moving with these constraints can be found in http://www.geometrylab.de/RacingRobots/

We introduce a new type of movement constraints for a swarm of robots in a grid environment inspired by Alex Randolphs board game Ricochet Robots. We assume that once a robot starts to drive in a certain direction, it does not stop its movement until it hits an obstacle wall or another robot. (This property can be used to model robots with very limited abilities for self-localization.) We show that the question whether a given cell can be reached is NP-hard for arbitrary environments. A Java applet for simulating robot swarms moving with these constraints can be found in http://www.geometrylab.de/RacingRobots/.

We introduce a new type of movement constraints for a swarm of robots in a grid environment. This type is inspired by Alex Randolphs board game Ricochet Robot and may be used to model robots with very limited abilities for self localization: We assume that once a robot starts to drive in a certain direction, it does not stop its movement until it hits an obstacle wall or another robot. We show that the question, whether a given cell can be reached is NP-complete for arbitrary environments. A Java applet for simulating robot swarms moving with these constraints can be found in http://www.geometrylab.de/RacingRobots/

We address a shortest path problem in a given uncapacited and undirected network N = (V,E) with positive edge costs. In addition we are given a single source-destination pair (s,t), a shortest pathst connecting s and t and a new edge e = (p,q) 6∈ �st. The task is to find a minimum number of edges Ec ⊆ E and the minimum weight increase for each edge ec ∈ Ec such that the shortest pathst between s and t traverses edge e. We show that the problem is NP-hard and give a heuristic scheme for the problem.

Abstract We introduce a type of movement,constraints for a swarm,of robots in a grid environment, which is inspired by Alex Randolph’s board game,Ricochet Robot and new,to the field of robot motion,planning. This type of movement,may,be used to model,robots with very limited abilities for self localization: We assume,that once a robot starts to drive in a certain direction, it does not stop its movement until it hits an obstacle wall or another,robot. We give some,lower bounds,on the number of robots needed to reach every cell. Especially, it is easy to see that three robots are necessary and sufficient to reach every cell in a simple rectangular environment. Further, we consider the question, whether,a certain cell can be reached,is in arbitrary environments. A Java applet for simulating robot swarms,moving,with these constraints can be found in http://www.geometrylab.de/RacingRobots/ Key words: Robot navigation, unknown environment, navigation error, robot swarms, NP-hardness, PSPACE-completeness. 1,Introduction Robot motion,planning has received a lot of attention both in computational