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一到达时，如何以在线方式确定最优设施集合并开启设施以服务这些请求。本文研究了该问题的两个变体：（1）带权值请求和（2）拥塞。这两个变体均源于现实应用场景的驱动，而此前已知的在线设施选址结果无法直接推广用于分析它们。带权值请求：在此变体中，每个需求请求是一个二元组$(x,w)$，其中$x$为需求的标准位置，$w$为请求的对应权值。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，对于$n$个请求，我们在秘书模型中提出了一种在线算法，其竞争比为$\mathcal{O}(\log n)$，并证明该算法是最优的。拥塞：拥塞变体考虑每个设施所服务的请求数量增加时产生的额外拥塞成本。对于该变体，当拥塞成本为单项式形式时，我们证明存在一种算法可以达到常数竞争比。该常数依赖于单项式的指数和设施开启成本，但与请求数量无关。