The classic online facility location problem deals with finding the optimal set of facilities in an online fashion when demand requests arrive one at a time and facilities need to be opened to service these requests. In this work, we study two variants of the online facility location problem; (1) weighted requests and (2) congestion. Both of these variants are motivated by their applications to real life scenarios and the previously known results on online facility location cannot be directly adapted to analyse them. Weighted requests: In this variant, each demand request is a pair $(x,w)$ where $x$ is the standard location of the demand while $w$ is the corresponding weight of the request. The cost of servicing request $(x,w)$ at facility $F$ is $w\cdot d(x,F)$. For this variant, given $n$ requests, we present an online algorithm attaining a competitive ratio of $\mathcal{O}(\log n)$ in the secretarial model for the weighted requests and show that it is optimal. Congestion: The congestion variant considers the case when there is an additional congestion cost that grows with the number of requests served by each facility. For this variant, when the congestion cost is a monomial, we show that there exists an algorithm attaining a constant competitive ratio. This constant is a function of the exponent of the monomial and the facility opening cost but independent of the number of requests.

翻译：经典的在线设施选址问题研究当需求请求逐个到达时，如何以在线方式确定最优的设施集合，并通过开设设施来服务这些请求。本文研究了在线设施选址问题的两个变体：（1）加权请求变体；（2）拥塞变体。这两种变体均源于实际应用场景的驱动，而现有关于在线设施选址问题的已知结果无法直接适用于对其进行分析。加权请求变体：在此变体中，每个需求请求是一个二元组$(x,w)$，其中$x$是需求的标准位置，$w$是请求对应的权重。在设施$F$处服务请求$(x,w)$的成本为$w\cdot d(x,F)$。针对该变体，给定$n$个请求，我们提出了一种在线算法，在秘书模型下针对加权请求实现了$\mathcal{O}(\log n)$的竞争比，并证明该算法具有最优性。拥塞变体：拥塞变体考虑了存在额外拥塞成本的情况，该成本随每个设施所服务的请求数量增加而增长。针对该变体，当拥塞成本为单项式时，我们证明存在一种能够实现常数竞争比的算法。该常数是单项式指数与设施开设成本的函数，但与请求数量无关。