The Price of Symmetric Line Plans in the Parametric City

TL;DR

This paper investigates the trade-offs between symmetric and optimal line plans in public transport within a parametric, symmetric city model, showing that symmetric plans are often nearly optimal and providing approximation algorithms.

Contribution

It introduces a formal model for line planning in a parametric city, analyzes the symmetry gap, and offers approximation algorithms with practical implications.

Findings

Symmetric line plans are nearly optimal in fixed-parameter scenarios.

The symmetry gap is small when a specific city parameter is fixed.

Symmetric plans are effective in most practical cases.

Abstract

We consider the line planning problem in public transport in the Parametric City, an idealized model that captures typical scenarios by a (small) number of parameters. The Parametric City is rotation symmetric, but optimal line plans are not always symmetric. This raises the question to quantify the symmetry gap between the best symmetric and the overall best solution. For our analysis, we formulate the line planning problem as a mixed integer linear program, that can be solved in polynomial time if the solutions are forced to be symmetric. The symmetry gap is provably small when a specific Parametric City parameter is fixed, and we give an approximation algorithm for line planning in the Parametric City in this case. While the symmetry gap can be arbitrarily large in general, we show that symmetric line plans are a good choice in most practical situations.