Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

Telebus Berlin: Vehicle Scheduling in a Dial-a-Ride System

Telebus is Berlin's dial-a-ride system for handicapped people who cannot use the public transportation system. The service is provided by a fleet of about 100 mini-buses and includes assistance in getting in and out of the vehicle. Telebus has between 1,000 and 1,500 transportation requests per day. The problem is to schedule these requests onto the vehicles such that punctual service is provided while operation costs are minimized. Addi tional constraints include pre-rented vehicles, fixed bus driver shift lengths, obligatory breaks, and different vehicle capacities. We use a set partitioning approach for the solution of the bus scheduling problem that consists of two steps. The first clustering step identifies segments of possible bus tours ("orders") such that more than one person is transported at a time; the aim in this step is to reduce the size of the problem and to make use of larger vehicle capacities. The problem of selecting a set of orders such that the traveling distance of the vehicles within the orders is minimal is a set partitioning problem that can be solved to optimality. I