We study the problem of efficiently and fairly allocating a set of indivisible goods among agents with identical and additive valuations for the goods. The objective is to maximize the Nash social welfare, which is the geometric mean of the agents’ valuations. While maximizing the Nash social welfare is NP-hard, a PTAS for this problem is presented by Nguyen and Rothe. The main contribution of this paper is to design a first additive PTAS for this problem, that is, we give a polynomial-time algorithm that maximizes the Nash social welfare within an additive error ευ_max, where ε is an arbitrary positive number and υ_max is the maximum utility of goods. The approximation guarantee of our algorithm is better than that of a PTAS. The idea of our algorithm is simple; we apply a preprocessing and then utilize an additive PTAS for the target load balancing problem given recently by Buchem et al. However, a nontrivial amount of work is required to evaluate the additive error of the output.

A transfer made outside ticket gates is called an out-of-station interchange (OSI). Tokyo Metro permits OSI between more than 20 pairs of stations, which we call OSI pairs, as of February 2024 for the purpose of making transfers more convenient and comfortable. In this study, we experimentally investigate how changes in OSI pairs affect congestion of the Tokyo Metro railway network during the morning rush hour, where passenger flows are simulated using Wardrop equilibrium based on real OD data. Our numerical results suggest that a minor modification to the current situation could lead to a 5% reduction in the number of links with a congestion rate exceeding 200%.