Approximating $k$-Median via Pseudo-Approximation




Shi Li

Monday, March 11, 2013
4:00pm 5130
Upson Hall



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 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 $\alpha=1+\sqrt{3}+\epsilon$.  Prior to our work, it was not even known whether opening $k + o(k)$ facilities would help improve the approximation ratio.