We prove a few new lower bounds on the randomized competitive ratio for the $k$-server problem and other related problems, resolving some long-standing conjectures. In particular, for metrical task systems (MTS) we asympotically settle the competitive ratio and obtain the first improvement to an existential lower bound since the introduction of the model 35 years ago (in 1987). More concretely, we show: 1. There exist $(k+1)$-point metric spaces in which the randomized competitive ratio for the $k$-server problem is $\Omega(\log^2 k)$. This refutes the folklore conjecture (which is known to hold in some families of metrics) that in all metric spaces with at least $k+1$ points, the competitive ratio is $\Theta(\log k)$. 2. Consequently, there exist $n$-point metric spaces in which the randomized competitive ratio for MTS is $\Omega(\log^2 n)$. This matches the upper bound that holds for all metrics. The previously best existential lower bound was $\Omega(\log n)$ (which was known to be tight for some families of metrics). 3. For all $k<n\in\mathbb N$, for *all* $n$-point metric spaces the randomized $k$-server competitive ratio is at least $\Omega(\log k)$, and consequently the randomized MTS competitive ratio is at least $\Omega(\log n)$. These universal lower bounds are asymptotically tight. The previous bounds were $\Omega(\log k/\log\log k)$ and $\Omega(\log n/\log \log n)$, respectively. 4. The randomized competitive ratio for the $w$-set metrical service systems problem, and its equivalent width-$w$ layered graph traversal problem, is $\Omega(w^2)$. This slightly improves the previous lower bound and matches the recently discovered upper bound. 5. Our results imply improved lower bounds for other problems like $k$-taxi, distributed paging and metric allocation. These lower bounds share a common thread, and other than the third bound, also a common construction.

翻译：我们证明了关于$k$-服务问题及相关问题随机化竞争比的一些新下界，解决了一些长期存在的猜想。具体而言，对于度量任务系统（MTS），我们渐近确定了其竞争比，并自该模型于35年前（1987年）提出以来首次改进了存在性下界。更具体地，我们证明：1. 存在$(k+1)$-点度量空间，其中$k$-服务问题的随机化竞争比为$\Omega(\log^2 k)$。这反驳了（已知在某些度量族中成立的）民间猜想——在所有至少包含$k+1$个点的度量空间中，竞争比为$\Theta(\log k)$。2. 由此可得，存在$n$-点度量空间，其中MTS的随机化竞争比为$\Omega(\log^2 n)$。这与所有度量空间的上界匹配。此前最佳存在性下界为$\Omega(\log n)$（已知在某些度量族中可达紧界）。3. 对所有满足$k<n\in\mathbb N$的整数，以及*所有*$n$-点度量空间，随机化$k$-服务竞争比至少为$\Omega(\log k)$，从而随机化MTS竞争比至少为$\Omega(\log n)$。这些普适下界在渐近意义下是紧的。此前相应下界分别为$\Omega(\log k/\log\log k)$和$\Omega(\log n/\log \log n)$。4. $w$-集度量服务系统问题（等价于宽度为$w$的层次图遍历问题）的随机化竞争比为$\Omega(w^2)$。这略微改进了先前的下界，并与最近发现的上界匹配。5. 我们的结果还改进了$k$-出租车、分布式分页及度量分配等其他问题的下界。这些下界具有共同的技术主线，且除第三个结论外均基于同一构造方法。