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∪Y. We would like to cover the clients by balls centered at
the servers. The objective function to minimize is the sum of the α-th
power of the radii of the balls. Here α≥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
3α and a cα 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
α 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 α≥log∣X∣.Comment: 19 page