  In this paper, we study the uniform capacitated $k$-median problem. Obtaining
a constant approximation algorithm for this problem is a notorious open
problem; most previous works gave constant approximations by either violating
the capacity constraints or the cardinality constraint. Notably, all these
algorithms are based on the natural LP-relaxation for the problem. The
LP-relaxation has unbounded integrality gap, even when we are allowed to
violate the capacity constraints or the cardinality constraint by a factor of
$2-\epsilon$.
  Our result is an $\exp(O(1/\epsilon^2))$-approximation algorithm for the
problem that violates the cardinality constraint by a factor of $1+\epsilon$.
This is already beyond the capability of the natural LP relaxation, as it has
unbounded integrality gap even if we are allowed to open $(2-\epsilon)k$
facilities. Indeed, our result is based on a novel LP for this problem.
  The version as we described is the hard-capacitated version of the problem,
as we can only open one facility at each location. This is as opposed to the
soft-capacitated version, in which we are allowed to open more than one
facilities at each location. We give a simple proof that in the uniform
capacitated case, the soft-capacitated version and the hard-capacitated version
are actually equivalent, up to a small constant loss in the approximation
ratio.
