The task of the train timetabling problem or track allocation problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. Especially for non-periodic instances models based on time expanded networks are often used. Unfortunately, the linear programming relaxation of these models is often extremely weak because these models do not describe combinatorial relations like overtaking possibilities very well. In this paper we extend the model by so called connected configuration subproblems. These subproblems perfectly describe feasible schedules of a small subset of trains (2-3) on consecutive track segments. In a Lagrangian relaxation approach we solve several of these subproblems together in order to produce solutions which consist of combinatorially compatible schedules along the track segments. The computational results on a mostly single track corridor taken from the INFORMS RAS Problem Solving Competition 2012 data indicate that our new solution approach is rather strong. Indeed, for this instance the solution of the Lagrangian relaxation is already integral.

The task of the train timetabling problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. This kind of problem has proven to be very challenging and numerous solution approaches have been proposed. One of the most successful approaches is based on time discretized network models. However, one of the major weaknesses of these models is that fractional solutions tend to change the order of trains along some track, which is not allowed for integer solutions, leading to poor relaxations. In this paper, we present an extension for these kind of models, which aims at overcoming these problems. By exploiting a configuration based formulation, we propose to extend the model with additional ordering constraints. These constraints enforce compatibility of orderings along a sequence of tracks and greatly improve the quality of the relaxations. We show in some promising preliminary computational experiments that our approach indeed helps to resolve many of the invalid overtaking problems of relaxations for the standard models.

In this paper we introduce an extension of the Traveling Salesman Problem (TSP), which is motivated by an important application in bioinformatics. In contrast to the TSP the costs do not only depend on each pair of two nodes traversed in succession in a cycle but on each triple of nodes traversed in succession. This problem can be formulated as optimizing a quadratic objective function over the traveling salesman polytope, so we call the combinatorial optimization problem quadratic TSP (QTSP). Besides its application in bioinformatics, the QTSP is a generalization of the Angular-Metric TSP and the TSP with reload costs. Apart from the TSP with quadratic cost structure we also consider the related Cycle Cover Problem with quadratic objective function (QCCP). In this work we present three exact solution approaches and several heuristics for the QTSP. The first exact approach is based on a polynomial transformation to a TSP, which is then solved by standard software. The second one is a branch-and-bound algorithm that relies on combinatorial bounds. The best exact algorithm is a branch-and-cut approach based on an integer programming formulation with problem-specific cutting planes. All heuristical approaches are extensions of classic heuristics for the TSP. Finally, we compare all algorithms on real-world instances from bioinformatics and on randomly generated instances. In these tests, the branch-and-cut approach turned out to be superior for solving the real-world instances from bioinformatics. Instances with up to 100 nodes could be solved to optimality in about ten minutes.

The aim of the train timetabling problem is to find a conflict free timetable for a set of passenger and freight trains along their routes in an infrastructure network. Several constraints like station capacities and train dependent running and headway times have to be satisfied. In this work we deal with large scale instances of the aperiodic train timetabling problem for the German railway network. The problem is modelled in a classical way via time discretised networks, its Lagrange-dual is solved by a bundle method. In order to handle the enormous number of variables and constraints dynamic graph generation and dynamic rolling horizon techniques are employed.