Line Planning in Public Transport: Bypassing Line Pool Generation

TL;DR

This paper investigates the computational complexity of line planning in public transport when all simple paths are considered as potential lines, revealing NP-hardness in general and polynomial solutions for specific cases.

Contribution

It demonstrates the NP-hardness of line planning with all simple paths and provides polynomial solutions for certain graph structures.

Findings

NP-hardness for general line planning with all simple paths

Polynomial solutions for trees and specific graph classes

No sub-linear approximation algorithms exist for the general problem

Abstract

Line planning, i.e. choosing paths which are operated by one vehicle end-to-end, is an important aspect of public transport planning. While there exists heuristic procedures for generating lines from scratch, most theoretical observations consider the problem of choosing lines from a predefined line pool. In this paper, we consider the complexity of the line planning problem when all simple paths can be used as lines. Depending on the cost structure, we show that the problem can be NP-hard even for paths and stars and that no polynomial time approximation of sub-linear performance is possible. Additionally, we identify polynomially solvable cases and present a pseudo-polynomial solution approach for trees.