Efficient Investment in Multi-Agent Models of Public Transportation

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.