Design Tasks and Their Complexity for Hybrid Level 3 of the European Train Control System

Railway networks have become increasingly important in recent times, especially to move freight and public transportation from road traffic and planes to more environmentally friendly trains. Since expanding the global railway network is time and resource consuming, maximizing the rail capacity on the existing infrastructure is desirable. However, simply running more trains is infeasible as certain constraints enforced by the train control system must be satisfied. The capacity of a network depends (amongst others) on the distance between trains allowed by this safety system. While most signaling systems rely on fixed blocks defined by costly hardware, new specifications provided by the ETCS Hybrid Level 3 (since recently also known as ETCS Level 2 with Hybrid Train Detection) allow the usage of virtual subsections. This additional degree of freedom allows for shorter train following times and, thus, more trains on existing railway tracks. On the other hand, new design tasks arise on which automated methods might be helpful for designers of modern railway networks. However, although first approaches exist that solve design problems arising within ETCS Hybrid Level 3, neither formal descriptions nor results on the computational complexity of the corresponding design tasks exist. In this paper, we fill this gap by providing a formal description of design tasks for the Hybrid Level 3 of the European Train Control System and proofs that these tasks are NP-complete or NP-hard, respectively. By that, we are providing a solid basis for the future development of methods to solve those tasks, which will be integrated into the Munich Train Control Toolkit available at https://github.com/cda-tum/mtct.