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：经典在线设施选址问题研究的是，在需求请求逐个到达且需开设设施以提供服务的情况下，以在线方式寻找最优设施集合。本文研究在线设施选址问题的两个变体：（1）加权请求与（2）拥塞。这两个变体均受其实际应用场景的驱动，而先前关于在线设施选址问题的已知结果无法直接用于分析它们。加权请求：在该变体中，每个需求请求是一个对 $(x,w)$，其中 $x$ 是需求的标准位置，$w$ 是请求的对应权重。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，我们在秘书模型中提出一种在线算法，其竞争比为 $\mathcal{O}(\log n)$，并证明该结果是最优的。拥塞：拥塞变体考虑另一种情况，即存在随每个设施服务的请求数量增加而增长的附加拥塞成本。针对该变体，当拥塞成本为单项式时，我们证明存在一种达到常数竞争比的算法。该常数取决于单项式的指数与设施开设成本，但与请求数量无关。