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The classic online facility location problem deals with finding the optimalset of facilities in an online fashion when demand requests arrive one at atime and facilities need to be opened to service these requests. In this work,we study two variants of the online facility location problem; (1) weightedrequests and (2) congestion. Both of these variants are motivated by theirapplications to real life scenarios and the previously known results on onlinefacility 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 correspondingweight 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 onlinealgorithm attaining a competitive ratio of $\mathcal{O}(\log n)$ in thesecretarial model for the weighted requests and show that it is optimal.  Congestion: The congestion variant considers the case when there is anadditional congestion cost that grows with the number of requests served byeach facility. For this variant, when the congestion cost is a monomial, weshow that there exists an algorithm attaining a constant competitive ratio.This constant is a function of the exponent of the monomial and the facilityopening cost but independent of the number of requests.

Online facility location with weights and congestion