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.

翻译：公共交通网络中的票价设定是一项重要挑战。一种常用方法是将网络划分为票价分区（"分区"），并根据行程穿越的分区数量确定价格（"定价"）。本文针对给定的凹定价函数，重点研究寻找分区问题的收益最优解。我们考虑具有$n$个顶点的树状网络，因为树结构本身已构成非平凡的算法挑战。主要研究成果是高效算法：一个提供简单的$\mathcal{O}(\log n)$近似解，另一个更复杂的算法提供$\mathcal{O}(\log n/\log \log n)$近似解。我们展示了如何在所有需求均源于同一出发点的根节点实例中精确求解该问题。对于路径网络，我们证明了问题的强NP困难性并概述了PTAS方案。此外，我们证明了针对多个自然问题参数，计算最优解属于FPT或XP复杂度类。