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23-22
We present incremental heuristics for the Periodic Event Scheduling Problem (PESP), the standard mathematical tool to optimize periodic timetables in public transport. The core of our method is to solve successively larger subinstances making use of previously found solutions. Introducing the technical notion of free stratifications, we formulate a general scheme for incremental heuristics for PESP. More practically, we use line and station information to create heuristics that add lines or stations one by one, and we evaluate these heuristics on instances of the benchmarking library PESPlib. This approach is indeed viable, and leads to new incumbent solutions for six PESPlib instances.
23-23
The Periodic Event Scheduling Problem (PESP) is a notoriously hard combinatorial optimization problem, essential for the design of periodic timetables in public transportation. The coefficients of the integer variables in the standard mixed integer linear programming formulations of PESP are the period time, e.g., 60 for a horizon of one hour with a resolution of one minute. In many application scenarios, lines with different frequencies have to be scheduled, leading to period times with many divisors. It then seems natural to consider derived instances, where the period time is a divisor of the original one, thereby smaller, and bounds are scaled and rounded accordingly. To this end, we identify two rounding schemes: wide and tight. We then discuss the approximation performance of both strategies, in theory and practice.
23-16
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.