Fair Transit Stop Placement: A Clustering Perspective and Beyond

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.

💡 Research Summary

The paper tackles the Transit Stop Placement (TrSP) problem in arbitrary metric spaces, where each agent travels between a source–destination pair and may either walk directly or use a shuttle service that connects two selected stops. The authors introduce a flexible model that distinguishes walking costs (metric d) from shuttle travel costs (metric d′), requiring only the triangle inequality. Fairness is examined through two lenses: justified representation (JR), a relaxation of core stability, and the core itself. Both notions are approximated: β‑JR requires that any deviating coalition cannot improve its members’ costs by more than a factor β, while (α, β)‑core strengthens the size requirement of a deviating group by α and limits its cost improvement by β.

A central contribution is establishing a structural correspondence between TrSP and centroid clustering. When shuttle travel times are negligible (d′≈d), TrSP reduces to a classic fair clustering problem. The authors prove that any clustering algorithm achieving ρ‑proportional fairness (PF) yields a (2, ρ)‑core solution for TrSP. Applying the Greedy Capture (GC) algorithm, which attains a (1+√2)‑PF guarantee, leads to a (2, 1+√2)‑core and a (2+√5)≈4.24‑JR approximation for TrSP. Conversely, a β‑JR algorithm for TrSP can be transformed into a 2β‑PF clustering solution, implying that in general metric spaces a perfect JR (and thus core) outcome may not exist. The paper proves a lower bound of 1.366 on the approximability of JR and shows that no clustering‑based method can beat a 3‑JR factor.

To surpass these limits, the authors propose the Expanding Cost Algorithm (ECA). Unlike clustering approaches that consider only distances, ECA evaluates pairs of stops and directly minimizes agents’ actual travel costs. ECA achieves a tight (1+√2)≈2.41‑JR approximation for any metric satisfying the triangle inequality, but it does not provide any bounded core guarantee. Recognizing the complementary strengths of GC (good core, weaker JR) and ECA (strong JR, no core), they introduce a parameterized λ‑Hybrid algorithm. By tuning λ≥0, the algorithm interpolates between singleton‑stop selection (GC) and paired‑stop selection (ECA). The analysis shows that λ‑Hybrid guarantees JR within a factor of (λ+3+√(λ²+10λ+9))/2 and a (2, √(λ²+6λ+1)+λ+1 / 2λ)‑core approximation. Thus, decision makers can trade off between fairness notions by adjusting λ.

The paper also presents experimental results on a small‑scale public car‑pooling dataset. The experiments compare GC‑TrSP, ECA, and λ‑Hybrid across JR and core metrics, confirming that λ‑Hybrid can balance both objectives depending on λ. Additionally, the authors prove that minimizing total cost for TrSP in general metric spaces is NP‑hard, justifying the focus on approximation algorithms.

Overall, the work extends fairness research in public transportation from line‑only settings to realistic metric spaces, bridges the gap between fair clustering and transit stop placement, and offers a suite of algorithms with provable guarantees and practical tunability. This advances both the theoretical understanding of fairness in location problems and provides actionable tools for planners seeking equitable transit infrastructure.