Linear Time Approximation Schemes for Geometric Maximum Coverage

Abstract: 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. We want to place m a * b rectangles such that the sum of the weights of the points in P covered by these rectangles is maximized.For any fixed > 0, we present efficient approximation schemes that can find (1-{\epsilon})-approximation to the optimal solution.In particular, for m = 1, our algorithm runs in linear time O(n log( 1/{\epsilon})), improving over the previous result. For m > 1, we present an algorithm that runs in O(n/{\epsilon}log(1/{\epsilon})+m(1/{\epsilon})^O(min(sqrt(m,1/{\epsilon}))) time.