We study the K-Facility Reallocation problem on the real line, where we maintain K facility locations over T stages, based on the stage-dependent locations of n 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 K-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 K-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 K-server problem, we present a constant-competitive algorithm for $K=2$ facilities.

我们在实线上研究K设施重新分配问题，在该阶段，我们根据n个代理的阶段相关位置来维护T阶段的K设施位置。每个代理在每个阶段都连接到最近的设施，并且设施可以从一个阶段移动到另一阶段，以适应不同的代理位置。目的是在所有阶段将代理的连接成本以及设施的总移动成本最小化。该ķ-设施重新分配问题由de Keijzer和Wojtczak提出，他们主要关注单一设施的特殊情况。使用基于LP的方法，我们提出了多项式时间算法，该算法可为任意数量的设施计算最佳解决方案。我们还考虑了在线K设施重新分配问题，该算法以逐步的方式了解代理位置。通过利用与经典K服务器问题的有趣联系，我们提出了一种恒定竞争算法$ķ=2$ 设备。