On the split closure of the periodic timetabling polytope
研究了周期时刻表优化中PESP问题的分裂闭包，证明分裂不等式与翻转不等式等价，并分析了其分离复杂度及不同公式化下的闭包强度，为PESPlib实例提供了更好的对偶界。
Abstract 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 $$\hbox {P}=\hbox {NP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>P</mml:mtext> <mml:mo>=</mml:mo> <mml:mtext>NP</mml:mtext> </mml:mrow> </mml:math> , 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.