Efficient Investment in Multi-Agent Models of Public Transportation

TL;DR

This paper analyzes multi-agent models for public transportation investment, revealing computational complexities and providing polynomial algorithms for specific cases, with implications for network design.

Contribution

It introduces new complexity results and algorithms for optimizing resource allocation in multi-agent public transportation models.

Findings

Polynomial-time solutions for certain single-agent network improvements

NP-completeness of egalitarian welfare optimization on line graphs

Implications for railway network design models

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.