Near-linear time approximation schemes for geometric maximum coverage

We study approximation algorithms for the following geometric version of the maximum coverage problem: Let P be a set of n weighted points in the plane. Let D represent a planar object, such as a rectangle, or a disk. We want to place m copies of D such that the sum of the weights of the points in P covered by these copies is maximized. For any fixed ε>0, we present efficient approximation schemes that can find a (1−ε)-approximation to the optimal solution. In particular, for m=1 and for the special case where D is a rectangle, our algorithm runs in time O(nlog⁡(1ε)), improving on the previous result. For m>1 and the rectangular case, our algorithm runs in O(nεlog⁡(1ε)+mεlog⁡m+m(1ε)O(min⁡(m,1ε))) time. For a more general class of shapes (including disks, polygons with O(1) edges), our algorithm runs in O(n(1ε)O(1)+mϵlog⁡m+m(1ε)O(min⁡(m,1ε2))) time.