On the Split Closure of the Periodic Timetabling Polytope

TL;DR

This paper explores the split closure of the periodic timetabling polytope in PESP, showing its equivalence to flip inequalities, analyzing their computational complexity, and demonstrating their effectiveness in improving dual bounds.

Contribution

It establishes the equivalence of split inequalities and flip inequalities, analyzes their separation complexity, and compares split closures across different formulations.

Findings

Split inequalities are identical to flip inequalities.

Separation of flip inequalities is pseudo-polynomial, but linear-time for fixed cycles.

Split cuts can significantly improve dual bounds on benchmark instances.

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 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.