The Second Chvatal Closure Can Yield Better Railway Timetables

We investigate the polyhedral structure of the Periodic Event Scheduling Problem (PESP), which is commonly used in periodic railway timetable optimization. This is the first investigation of Chvátal closures and of the Chvátal rank of PESP instances. In most detail, we first provide a PESP instance on only two events, whose Chvátal rank is very large. Second, we identify an instance for which we prove that it is feasible over the first Chvátal closure, and also feasible for another prominent class of known valid inequalities, which we reveal to live in much larger Chvátal closures. In contrast, this instance turns out to be infeasible already over the second Chvátal closure. We obtain the latter result by introducing new valid inequalities for the PESP, the multi-circuit cuts. In the past, for other classes of valid inequalities for the PESP, it had been observed that these do not have any effect in practical computations. In contrast, the new multi-circuit cuts that we are introducing here indeed show some effect in the computations that we perform on several real-world instances – a positive effect, in most of the cases.