Linear Time Approximation Schemes for Geometric Maximum Coverage

TL;DR

This paper introduces efficient approximation algorithms for the geometric maximum coverage problem, achieving near-optimal solutions in linear or near-linear time for fixed parameters, improving computational efficiency over previous methods.

Contribution

The paper develops linear and near-linear time approximation schemes for geometric maximum coverage, with improved algorithms for both single and multiple rectangle placements.

Findings

For m=1, the algorithm runs in O(n log(1/ε)) time.

For m>1, the algorithm runs in O(n/ε log(1/ε) + m(1/ε)^{O(min(√m,1/ε))}) time.

The schemes achieve (1-ε)-approximation efficiently for fixed ε.

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.