Fair Transit Stop Placement: A Clustering Perspective and Beyond

TL;DR

This paper studies the Transit Stop Placement Problem (TrSP) in general metric spaces with fairness objectives, introducing beta-JR and (alpha,beta)-core as relaxations and revealing deep connections to fair clustering via ρ-Proportional Fairness. It proves a meta-result that ρ-PF solutions in the induced clustering instance yield (2,ρ)-core guarantees for TrSP, and shows hardness barriers for JR existence and constant-factor JR via lower bounds (≈1.366) and (3)-JR impossibility for clustering. To overcome these limits, the Expanding Cost Algorithm (ECA) achieves a tight (1+√2) ≈ 2.414-JR under arbitrary transit costs but lacks any core guarantee, while GC-TrSP provides a constant-factor core guarantee through PF-based reductions. A tunable λ-Hybrid algorithm balances JR and core guarantees, delivering explicit (JR) and core bounds that interpolate between ECA and GC-TrSP, with empirical evaluation on real small-market carpooling data confirming the practical viability of the proposed approach. Overall, the work advances fair stop-placement design by combining rigorous fairness guarantees with flexible cost modeling and empirical validation, laying groundwork for scalable, equitable transit planning in general metric spaces.

Abstract

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.

Figures (10)

• Figure 1: Transit stop placement example. Each travel route connects an agent pair $(a_i, b_i)$, with six agents in total (blue circles). There are four candidate stop locations, $c_1$ to $c_4$ (red squares). Panels (a) and (b) illustrate two different placement choices, with the selected stops marked by yellow stars. Red arrows indicate the shuttle transit routes.
• Figure 2: Overview of JR approximation ratios. The two shaded regions with diagonal lines indicate the lower bounds for general algorithms and clustering algorithms, respectively. The points at $2.414$ and $4.236$ correspond to the ECA and GC-TrSP algorithms. The performance of the $\lambda$-Hybrid algorithm ranges between $3$ and $4.236$, depending on the choice of the parameter $\lambda$.
• Figure 3: Graphical representation of clustering instance $\mathcal{I}^C$. Each edge in the graph has unit length $1$ and distances between pairs of points are given by the shortest path between them (infinite distance if the pair is not connected). Datapoints which are candidate centers are labeled by blue rectangles.
• Figure 4: One edge in the complete graph $K_z$ with endpoint vertices $1$ and $2$
• Figure 5: Parameter $\lambda \in [0,1]$. Red solid line represents the JR approximation ratio of $\frac{\lambda+3+\sqrt{\lambda^2+10\lambda+9}}{2}$ and blue dashed line represents the parameterized function $\frac{\sqrt{\lambda^2 + 6\lambda + 1} + \lambda + 1}{2\lambda}$ of core approximation.

Theorems & Definitions (45)

• Proposition 1
• Definition 1: $\beta$-Justified Representation ($\beta$-JR)
• Definition 2: $(\alpha,\beta)$-core
• Definition 3: $\rho$-Proportional Fairness ($\rho$-PF)
• Theorem 1
• proof
• Proposition 2
• Theorem 2
• proof
• Lemma 1