Constrained Distributed Heterogeneous Two-Facility Location Problems with Max-Variant Cost

We study a constrained distributed heterogeneous two-facility location problem, where a set of agents with private locations on the real line are divided into disjoint groups. The constraint means that the facilities can only be built in a given multiset of candidate locations and at most one facility can be built at each candidate location. Given the locations of the two facilities, the cost of an agent is the distance from her location to the farthest facility (referred to as max-variant). Our goal is to design strategyproof distributed mechanisms that can incentivize all agents to truthfully report their locations and approximately optimize some social objective. A distributed mechanism consists of two steps: for each group, the mechanism chooses two candidate locations as the representatives of the group based only on the locations reported by agents therein; then, it outputs two facility locations among all the representatives. We focus on a class of deterministic strategyproof distributed mechanisms and analyze upper and lower bounds on the distortion under the Average-of-Average cost (average of the average individual cost of agents in each group), the Max-of-Max cost (maximum individual cost among all agents), the Average-of-Max cost (average of the maximum individual cost among all agents in each group) and the Max-of-Average cost (maximum of the average individual cost of all agents in each group). Under four social objectives, we obtain constant upper and lower distortion bounds.