The Periodic Event Scheduling Problem (PESP) is the central mathematical tool for periodic timetable optimization in public transport. PESP can be formulated in several ways as a mixed-integer linear program with typically general integer variables. We investigate the split closure of these formulations and show that split inequalities are identical with the recently introduced flip inequalities. While split inequalities are a general mixed-integer programming technique, flip inequalities are defined in purely combinatorial terms, namely cycles and arc sets of the digraph underlying the PESP instance. It is known that flip inequalities can be separated in pseudo-polynomial time. We prove that this is best possible unless P $=$ NP, but also observe that the complexity becomes linear-time if the cycle defining the flip inequality is fixed. Moreover, introducing mixed-integer-compatible maps, we compare the split closures of different formulations, and show that reformulation or binarization by subdivision do not lead to stronger split closures. Finally, we estimate computationally how much of the optimality gap of the instances of the benchmark library PESPlib can be closed exclusively by split cuts, and provide better dual bounds for five instances.

翻译：周期事件调度问题（PESP）是公共交通中周期时间表优化的核心数学工具。PESP可通过多种方式表述为混合整数线性规划，通常包含一般整数变量。本文研究了这些表述的分割闭包，并证明分割不等式与近期引入的翻转不等式等价。尽管分割不等式是一类通用的混合整数规划技术，但翻转不等式完全以组合术语定义，即PESP实例所对应有向图中的圈与弧集。已知翻转不等式可在伪多项式时间内分离。我们证明除非P=NP，否则这是最优复杂度，但同时也观察到若定义翻转不等式的圈固定，则复杂度可降为线性时间。此外，通过引入混合整数兼容映射，我们比较了不同表述的分割闭包，证明通过细分进行重构或二值化不会增强分割闭包。最后，我们通过计算评估基准库PESPlib中实例的最优性间隙能被分割割闭包单独弥补的程度，并为五个实例提供了更优的对偶界。