We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. [SODA' 07]). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of $k$ mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics. A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the $k$-server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.

翻译：我们提出了移动设施选址（MFL）问题的一种在线变体（由Demaine等人在[SODA'07]中提出），将其命名为在线移动设施选址（OMFL）问题。在OMFL问题中，初始给定一组包含$k$个移动设施的起始位置。请求逐一加入，每加入一个请求后，可在下一个请求到达前对设施位置进行部分调整。每个请求始终被分配给最近的设施，该分配成本为请求到设施的距离。目标是最小化总成本，该总成本由设施重定位成本与请求到最近设施的距离成本构成。我们给出了OMFL问题的下界，该下界即使在均匀度量下仍然成立。解决一般度量空间下OMFL问题的一种自然方法是利用层次化良好分隔树（HST）并直接求解HST上的OMFL问题。本文通过解决均匀度量下OMFL问题的广义变体（称为G-OMFL）迈出了该方向的第一步。我们设计了一种简单的确定性在线算法，并给出了该算法的紧致分析。第二步仍然是一个开放问题。受$k$-服务商问题的启发，我们引入了OMFL问题的一种仅关注最小化移动成本的新变体，称为M-OMFL。此外，我们给出了M-OMFL的下界，该下界甚至在均匀度量下同样适用。