The weighted $k$-server problem is a natural generalization of the $k$-server problem in which the cost incurred in moving a server is the distance traveled times the weight of the server. Even after almost three decades since the seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem remains poorly understood, even on the simplest class of metric spaces -- the uniform metric spaces. In particular, in the case of randomized algorithms against the oblivious adversary, neither a better upper bound that the doubly exponential deterministic upper bound, nor a better lower bound than the logarithmic lower bound of unweighted $k$-server, is known. In this article, we make significant progress towards understanding the randomized competitive ratio of weighted $k$-server on uniform metrics. We cut down the triply exponential gap between the upper and lower bound to a singly exponential gap by proving that the competitive ratio is at least exponential in $k$, substantially improving on the previously known lower bound of about $\ln k$.

翻译：加权美元服务器问题是一个自然的典型问题,即移动服务器的费用是服务器重量的距离乘以距离。即使在Fiat和Ricklin的开创性工作(1994年)近三十年后,这个问题的竞争比率仍然不甚为人理解,甚至在最简单的公用空间类别 -- -- 统一的公用空间上也是如此。特别是,在对不明对手随机化算法的情况下,既未比二重指数确定性上限约束值高,也未比未加权美元服务器的对数较低约束值低,两者的相对约束值要好得多。在本篇文章中,我们在了解统一的公用计量标准加权美元服务器的随机竞争比率方面取得了重大进展。我们缩小了上下层和下层之间的三重指数差距,通过证明竞争性比率至少以美元计为指数,从而缩小了上层和下层之间的微指数差距,从而大大改进了先前已知的约美元的较低约束值。