A Compact Cycle Formulation for the Multiperiodic Event Scheduling Problem

TL;DR

This paper introduces a new cycle-based formulation for the Multiperiodic Event Scheduling Problem, enabling efficient and exact solutions for heterogeneous public transport timetabling, outperforming previous models.

Contribution

It extends the strongest known PESP formulation to MPESP, using a cycle basis derived from a specialized spanning tree, and proves a multiperiodic cycle periodicity property.

Findings

Successfully solves large-scale real-world instances.

Dramatically outperforms existing arc-based models.

Achieves optimality or small gaps in nearly all tested cases.

Abstract

The Periodic Event Scheduling Problem (PESP) is a fundamental model in periodic timetabling for public transport systems, assuming a common period across all events. However, real-world networks often feature heterogeneous service frequencies. This paper studies the Multiperiodic Event Scheduling Problem (MPESP), a generalization of PESP that allows each event to recur at its own individual period. While more expressive, MPESP presents new modeling challenges due to the loss of a global period. We present a cycle-based formulation for MPESP that extends the strongest known formulation for PESP and, in contrast to existing approaches, is valid for any MPESP instance. Crucially, the formulation requires a cycle basis derived from a spanning tree satisfying specific structural properties, which we formalize and algorithmically construct, extending the concept of sharp spanning trees to rooted instances. We further prove a multiperiodic analogue of the cycle periodicity property. Our new formulation solves nearly all tested instances, including several large-scale real-world public transport networks, to optimality or with small optimality gaps, dramatically outperforming existing arc-based models. The results demonstrate the practical potential of MPESP in capturing heterogeneous frequencies without resorting to artificial event duplication.