Fare Zone Assignment on Trees

TL;DR

This paper develops efficient algorithms for fare zone partitioning in tree networks to maximize revenue, providing approximation guarantees and complexity results for various special cases.

Contribution

It introduces approximation algorithms with logarithmic factors, exact solutions for rooted cases, and hardness results for general instances in fare zoning on trees.

Findings

O(log n)-approximation algorithm for general trees

Exact solutions for rooted instances with demand at a single source

APX-hardness and NP-hardness results for specific graph classes

Abstract

Designing fare systems for public transportation networks is a challenging task. A popular approach is to partition the network into fare zones (``zoning'') and fix journey prices depending on the number of traversed zones (``pricing''). In this paper, we focus on finding revenue-optimal solutions to the zoning problem for a given subadditive pricing function. We consider tree networks with $n$ vertices, since trees already pose non-trivial algorithmic challenges. Our main results are efficient algorithms that yield a simple $\mathcal{O}(\log n)$-approximation as well as a more involved $\mathcal{O}(\log n/\log \log n)$-approxi\-ma\-tion. We show that rooted instances, in which all demand arises at a single source, can be solved exactly. We further show APX-hardness for general instances on star graphs. For paths, we prove strong NP-hardness and outline a PTAS. Moreover, we show that computing an optimal solution is in FPT or XP for several natural problem parameters.