Reallocating multiple facilities on the line

We study the -Facility Reallocation problem on the real line, where we maintain facility locations over stages, based on the stage-dependent locations of agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. The -Facility Reallocation problem was introduced by de Keijzer and Wojtczak, where they mostly focused on the special case of a single facility. Using an LP-based approach, we present a polynomial time algorithm that computes the optimal solution for any number of facilities. We also consider the online -Facility Reallocation problem, where the algorithm becomes aware of agent locations in a stage-by-stage fashion. By exploiting an interesting connection to the classical -server problem, we present a constant-competitive algorithm for facilities.