Facility Location problems ask to place facilities in a way that optimizes a given objective function so as to provide a service to all clients. These are one of the most well-studied optimization problems spanning many research areas such as operations research, computer science, and management science. Traditionally, these problems are solved with the assumption that clients need to be served by one facility each. In many real-world scenarios, it is very likely that clients need a robust service that requires more than one facility for each client. In this paper, we capture this robustness by exploring a generalization of Facility Location problems, called Multi-Facility Location problems, in the online setting. An additional parameter k, which represents the number of facilities required to serve a client, is given. We propose the first online algorithms for the metric and non-metric variants of Multi-Facility Location and measure their performance with competitive analysis, the standard to measure online algorithms, in the worst case, in which the cost of the online algorithm is compared to that of the optimal offline algorithm that knows the entire input sequence in advance.

中文翻译：

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在线多设施位置

设施位置问题要求以优化给定目标函数的方式放置设施，以便为所有客户提供服务。这些是跨许多研究领域（如运筹学、计算机科学和管理科学）研究得最充分的优化问题之一。传统上，这些问题是通过假设每个客户需要由一个设施服务来解决的。在许多实际场景中，客户很可能需要一个强大的服务，每个客户需要一个以上的设施。在本文中，我们通过探索在线设置中的设施位置问题（称为多设施位置问题）的泛化来捕捉这种稳健性。给出了一个附加参数 k，它表示为客户提供服务所需的设施数量。