An improved approximation algorithm for k-Median

We give a polynomial-time approximation algorithm for the (not necessarily metric) $k$-Median problem. The algorithm is an $\alpha$-size-approximation algorithm for $\alpha < 1 + 2 \ln(n/k)$. That is, it guarantees a solution having size at most $\alpha\times k$, and cost at most the cost of any size-$k$ solution. This is the first polynomial-time approximation algorithm to match the well-known bounds of $H_\Delta$ and $1 + \ln(n/k)$ for unweighted Set Cover (a special case) within a constant factor. It matches these bounds within a factor of 2. The algorithm runs in time $O(k m \log(n/k) \log m)$, where $n$ is the number of customers and $m$ is the instance size.