Covering many or few points with unit disks

Let P be a set of n weighted points. We study approximation algorithms for the following two continuous facility-location problems. In the first problem we want to place m unit disks, for a given constant m=1, such that the total weight of the points from P inside the union of the disks is maximized. We present a deterministic algorithm that can compute, for any e>0, a (1-e)-approximation to the optimal solution in O(n logn + e-4mlog2m (1/e)) time. In the second problem we want to place a single disk with center in a given constant-complexity region X such that the total weight of the points from P inside the disk is minimized. Here we present an algorithm that can compute, for any e>0, with high probability a (1+e)-approximation to the optimal solution in O(n (log3 n + e-4 log2 n )) expected time