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.

翻译：公共交通网络票制设计是一项具有挑战性的任务。常见方法是将网络划分为票价分区（"分区"），并根据乘客穿越的分区数量确定行程票价（"定价"）。本文聚焦于在给定次加性定价函数下，寻找使得收益最大化的分区问题最优解。我们以含$n$个顶点的树状网络为研究对象，因为树结构本身已蕴含显著算法挑战。主要成果包括：提出能实现简单$\mathcal{O}(\log n)$近似比的高效算法，以及更精细的$\mathcal{O}(\log n/\log \log n)$近似算法。我们证明当需求集中于单一源节点时的根化实例可精确求解，并进一步证明星形图上一般实例的APX难度。针对路径结构，我们证明强NP困难性并给出多项式时间近似方案（PTAS）。此外，研究表明针对多个自然问题参数，计算最优解属于固定参数可解类（FPT）或切片多项式复杂性类（XP）。