Approximating k-median via pseudo-approximation

We present a novel approximation algorithm for k-median that achieves an approximation guarantee of 1+√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 α-approximation algorithm for k-median, it is sufficient to give a pseudo-approximation algorithm that finds an α-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 α= 1+√3+ε. Prior to our work, it was not even known whether opening k + o(k) facilities would help improve the approximation ratio.