We study the transit stop placement (TrSP) problem in general metric spaces, where agents travel between source-destination pairs and may either walk directly or utilize a shuttle service via selected transit stops. We investigate fairness in TrSP through the lens of justified representation (JR) and the core, and uncover a structural correspondence with fair clustering. Specifically, we show that a constant-factor approximation to proportional fairness in clustering can be used to guarantee a constant-factor biparameterized approximation to core. We establish a lower bound of 1.366 on the approximability of JR, and moreover show that no clustering algorithm can approximate JR within a factor better than 3. Going beyond clustering, we propose the Expanding Cost Algorithm, which achieves a tight 2.414-approximation for JR, but does not give any bounded core guarantee. In light of this, we introduce a parameterized algorithm that interpolates between these approaches, and enables a tunable trade-off between JR and core. Finally, we complement our results with an experimental analysis using small-market public carpooling data.

翻译：我们在一般度量空间中研究公交站点布局问题，其中智能体在源-目的地对之间出行，可选择直接步行或通过选定公交站点使用班车服务。我们通过合理代表性和核心解的视角探讨公交站点布局中的公平性问题，并揭示了其与公平聚类之间的结构对应关系。具体而言，我们证明聚类中比例公平的常数因子近似可用来保证核心解的双参数化常数因子近似。我们建立了合理代表性1.366的近似下界，并进一步证明任何聚类算法对合理代表性的近似比不可能优于3。超越聚类方法，我们提出扩展成本算法，该算法对合理代表性实现了紧致的2.414近似，但无法提供有界核心解保证。基于此，我们提出一种参数化算法，能在不同方法间进行插值，并实现合理代表性与核心解之间的可调节权衡。最后，我们使用小规模市场拼车数据通过实验分析对理论结果进行了补充验证。