  For a given set of points in a metric space and an integer $k$, we seek to
partition the given points into $k$ clusters. For each computed cluster, one
typically defines one point as the center of the cluster. A natural objective
is to minimize the sum of the cluster center's radii, where we assign the
smallest radius $r$ to each center such that each point in the cluster is at a
distance of at most $r$ from the center. The best-known polynomial time
approximation ratio for this problem is $3.389$. In the setting with outliers,
i.e., we are given an integer $m$ and allow up to $m$ points that are not in
any cluster, the best-known approximation factor is $12.365$.
  In this paper, we improve both approximation ratios to $3+\epsilon$. Our
algorithms are primal-dual algorithms that use fundamentally new ideas to
compute solutions and to guarantee the claimed approximation ratios. For
example, we replace the classical binary search to find the best value of a
Lagrangian multiplier $\lambda$ by a primal-dual routine in which $\lambda$ is
a variable that is raised. Also, we show that for each connected component due
to almost tight dual constraints, we can find one single cluster that covers
all its points and we bound its cost via a new primal-dual analysis. We remark
that our approximation factor of $3+\epsilon$ is a natural limit for the known
approaches in the literature.
  Then, we extend our results to the setting of lower bounds. There are
algorithms known for the case that for each point $i$ there is a lower bound
$L_{i}$, stating that we need to assign at least $L_{i}$ clients to $i$ if $i$
is a cluster center. For this setting, there is a $ 3.83$ approximation if
outliers are not allowed and a ${12.365}$-approximation with outliers. We
improve both ratios to $3.5 + \epsilon$ and, at the same time, generalize the
type of allowed lower bounds.
