The tropical and zonotopal geometry of periodic timetables

TL;DR

This paper explores the geometric structure of periodic timetabling problems, revealing connections to tropical and zonotopal geometry, and introduces new heuristics and bounds for optimizing public transport schedules.

Contribution

It establishes novel geometric insights into the space of feasible timetables and cycle offsets, linking them to tropical and zonotopal geometry, and proposes new optimization heuristics.

Findings

Decomposition of timetable space into polytropes.

Identification of the cycle offset space as a zonotope.

New lower bounds on integral cycle basis width.

Abstract

The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.