  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.
