Efficient Investment in Multi-Agent Models of Public Transportation

TL;DR

This work studies two stylized public-transport investment problems under fairness vs efficiency: the Path Transit Problem (PTP) on a line and the Network Transit Problem (NTP) on a weighted graph. It shows a sharp complexity dichotomy: utilitarian objectives are tractable for PTP and for small numbers of agents in NTP, while egalitarian objectives rapidly become intractable, and the general NTP with many agents is NP-hard with strong inapproximability results. The authors introduce a budgeted discount framework, devise a modified Dijkstra algorithm for the single-agent NTP, extend it to two agents, and prove NP-hardness and inapproximability for the multi-agent case (via Set Cover reductions and connections to railway design). Their findings highlight fundamental computational barriers to achieving equitable transport improvements in large-scale networks, motivating further research into restricted domains and approximation techniques.

Abstract

We study two stylized, multi-agent models aimed at investing a limited, indivisible resource in public transportation. In the first model, we face the decision of which potential stops to open along a (e.g., bus) path, given agents' travel demands. While it is known that utilitarian optimal solutions can be identified in polynomial time, we find that computing approximately optimal solutions with respect to egalitarian welfare is NP-complete. This is surprising as we operate on the simple topology of a line graph. In the second model, agents navigate a more complex network modeled by a weighted graph where edge weights represent distances. We face the decision of improving travel time along a fixed number of edges. We provide a polynomial-time algorithm that combines Dijkstra's algorithm with a dynamical program to find the optimal decision for one or two agents. By contrast, if the number of agents is variable, we find \np-completeness and inapproximability results for utilitarian and egalitarian welfare. Moreover, we demonstrate implications of our results for a related model of railway network design.

Figures (10)

• Figure 1: Illustration of the path transit problem. The upper panel displays agent routes, potential bus stops, and walking cost, while the lower panel presents the solution $S_1 = \{1,5\}$ (marked in orange) with its corresponding travel costs.
• Figure 2: Illustration of the network transit problem. The upper panel shows the original graph, the left panel depicts the shortest paths used by walking agents, and the right panel demonstrates how reducing the cost on the dashed edge alters these routes.
• Figure 3: Inferiority of terminal selection (left) compared to nonterminal selection (right) for the egalitarian objective.
• Figure 4: Illustration of instances constructed in \ref{['prop:gdy']}. The greedy algorithms yield an arbitrarily suboptimal solution.
• Figure 5: Instance for illustrating the execution of the algorithm. We display the shortest paths without (left) and with optimal (right) investment.

Theorems & Definitions (28)

• example 1
• example 2
• theorem 4
• theorem 5
• example 6
• proposition 1
• proof : Proof sketch
• theorem 7
• example 8
• theorem 9