Approximating $k$-Median via Pseudo-Approximation

Abstract

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our improved approximation ratio is achieved by exploiting the power of pseudo-approximation. More specifically, 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 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 significantly smaller cost than that of opening only $k$ facilities. Second, we give such a pseudo-approximation algorithm with $\alpha=1+\sqrt{3}+\epsilon$. Prior to our work, it was not even known whether opening $k+o(k)$ facilities would help improve the approximation ratio.