Approximate Clustering via Metric Partitioning

In this paper we consider two metric covering/clustering problems - textit{Minimum Cost Covering Problem} (MCC) and $k$-clustering. In the MCC problem, we are given two point sets $X$ (clients) and $Y$ (servers), and a metric on $X cup Y$. We would like to cover the clients by balls centered at the servers. The objective function to minimize is the sum of the $alpha$-th power of the radii of the balls. Here $alpha geq 1$ is a parameter of the problem (but not of a problem instance). MCC is closely related to the $k$-clustering problem. The main difference between $k$-clustering and MCC is that in $k$-clustering one needs to select $k$ balls to cover the clients. For any $eps > 0$, we describe quasi-polynomial time $(1 + eps)$ approximation algorithms for both of the problems. However, in case of $k$-clustering the algorithm uses $(1 + eps)k$ balls. Prior to our work, a $3^{alpha}$ and a ${c}^{alpha}$ approximation were achieved by polynomial-time algorithms for MCC and $k$-clustering, respectively, where $c > 1$ is an absolute constant. These two problems are thus interesting examples of metric covering/clustering problems that admit $(1 + eps)$-approximation (using $(1+eps)k$ balls in case of $k$-clustering), if one is willing to settle for quasi-polynomial time. In contrast, for the variant of MCC where $alpha$ is part of the input, we show under standard assumptions that no polynomial time algorithm can achieve an approximation factor better than $O(log |X|)$ for $alpha geq log |X|$.