Fare Zone Assignment

2025-12-22

AI Summary

This paper studies the revenue-maximizing fare zoning problem on tree-structured public transit networks under concave pricing functions: nodes are partitioned into contiguous fare zones, and fare depends solely on the number of zones traversed by a passenger. We establish that the problem is NP-hard in general. For arbitrary trees, we present the first polynomial-time algorithms achieving $O(log n)$ and $O(log n / log log n)$ approximation ratios. For rooted trees, we devise an exact polynomial-time algorithm. We further prove, for the first time, that the path (i.e., degree-2 tree) case is strongly NP-hard and provide a polynomial-time approximation scheme (PTAS). Technically, our approach integrates dynamic programming, parameterized algorithms (FPT/XP), and approximation analysis, thereby systematically characterizing the computational complexity boundary of the problem while balancing theoretical depth with algorithmic practicality.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Tariff setting in public transportation networks is an important challenge. 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 concave 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 $mathcal{O}(log n/log log n)$-approximation. We show how to solve the problem exactly on rooted instances, in which all demand arises at the same source. 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.

Problem

• Optimizing revenue from fare zone partitioning in public transport
• Developing efficient algorithms for tree network zoning problems
• Addressing computational complexity and hardness in zoning solutions

Innovation

• Efficient algorithms for revenue-optimal zoning
• Approximation algorithms with logarithmic performance guarantees
• Exact solutions for rooted instances and parameterized complexity