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 ma parts per thousand yen1, such that the total weight of the points from P inside the union of the disks is maximized. We present algorithms that compute, for any fixed epsilon > 0, a (1-epsilon)-approximation to the optimal solution in O(nlog n) 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 computes, for any fixed epsilon > 0, in O(nlog (2) n) expected time a disk that is, with high probability, a (1+epsilon)-approximation to the optimal solution.