Two-Facility Location Games with a Minimum Distance Requirement on a Circle.

We consider the games of locating two facilities with a minimum distance requirement, which was first introduced by Duan et al. 2019. In the setting, a mechanism maps the locations reported by strategic agents to two facilities, and the distance between them is at least d. The cost or utility of an agent is the sum of his distances to two facilities given that both facilities are favorite or obnoxious. One aims at designing strategyproof mechanisms, meanwhile achieving good approximation bounds on minimizing the total cost/maximizing the total utility or minimizing the maximum cost/maximizing the minimum utility. This paper is mainly focused on a circle network. We devise optimal strategyproof mechanisms for minimizing the maximum cost and maximizing the minimum utility, respectively. A group strategyproof mechanism with approximation ratio of 1/(2d) is designed for minimizing the total cost. And for maximizing the total utility, we establish a group strategyproof (2 - 2d)-approximation mechanism. We also revisit the line interval, while we propose a strategyproof mechanism towards maximizing the total utility, improving upon the previous bounds.