Approximating $k$-Median via Pseudo-Approximation

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+ϵ$, improving upon the decade-old ratio of $3+ϵ$. Our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an $\alpha$-approximation algorithm for $k$-median, it is sufficient to give a \emph{pseudo-approximation algorithm} that finds an $\alpha$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significant smaller cost than if only $k$ facilities were opened. Second, we give such a pseudo-approximation algorithm with $\alpha= 1+\sqrt{3}+\epsilon$\mathrm{.}Priortoourwork,itwasnotevenknownwhetheropening$k + o(k)$ facilities would help improve the approximation ratio.