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.

翻译：我们研究了两种旨在将有限且不可分割的资源投资于公共交通的简化多智能体模型。在第一个模型中，我们面临在给定智能体出行需求的情况下，沿（例如公交）线路决定开放哪些潜在站点的决策。虽然已知功利主义最优解可在多项式时间内确定，但我们发现计算基于平等主义福利的近似最优解是NP完全的。这一结论令人惊讶，因为我们的研究基于简单的线图拓扑结构。在第二个模型中，智能体在由加权图建模的更复杂网络中导航，其中边权重表示距离。我们面临沿固定数量的边改进旅行时间的决策。我们提出了一种结合Dijkstra算法与动态规划的多项式时间算法，可为一个或两个智能体找到最优决策。相比之下，当智能体数量可变时，我们发现了功利主义和平等主义福利的NP完全性与不可近似性结果。此外，我们论证了这些结果对相关铁路网络设计模型的启示。