In this paper, we study the weighted $k$-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) $k$-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted $k$-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use $c$-resource augmentation for $c < 2$. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least $\ell$ resource augmentation, where $\ell$ is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of $(2+\epsilon)\ell$ for any constant $\epsilon > 0$. In the online setting, an $\exp(k)$ lower bound is known for the competitive ratio of any randomized algorithm for the weighted $k$-server problem on the uniform metric. In contrast, we show that $2\ell$-resource augmentation can bring the competitive ratio down by an exponential factor to only $O(\ell^2 \log \ell)$. Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online.

翻译：本文研究均匀度量下带权k-服务问题在离线与在线两种场景中的表现。首先考虑离线场景。与可通过最小费用流在多项式时间内求解的（无权重）k-服务问题不同，即使在均匀度量上，带权k-服务问题也存在显著的计算复杂度下界。具体而言，我们证明：假设唯一游戏猜想成立，即使允许c<2的资源扩充，也不存在具有次多项式近似比的多项式时间算法。此外，若考虑该问题的自然线性规划松弛，要获得有界整性间隙至少需要ℓ单位的资源扩充，其中ℓ为不同服务权重的个数。作为补充，我们通过线性规划舍入方法得到常数近似算法，该算法对任意常数ε>0需(2+ε)ℓ单位的资源扩充。在线场景下，已知均匀度量上带权k-服务问题的任何随机化算法竞争比下界为exp(k)。相比之下，我们证明2ℓ单位的资源扩充可将竞争比指数级降低至仅O(ℓ² log ℓ)。该在线算法采用两阶段方法：首先通过在线原始-对偶框架获得分数解，随后对其进行在线舍入。