Linear Time Approximation Schemes for Geometric Maximum Coverage

Abstract

AbstractWe study approximation algorithms for the following geometric version of the maximum coverage problem: Let \(\mathcal {P}\) be a set of n weighted points in the plane. We want to place m \(a \times b\) rectangles such that the sum of the weights of the points in \(\mathcal {P}\) covered by these rectangles is maximized. For any fixed \(\varepsilon >0\), we present efficient approximation schemes that can find a \((1-\varepsilon )\)-approximation to the optimal solution. In particular, for \(m=1\), our algorithm runs in linear time \(O(n\log (\frac{1}{\varepsilon }))\), improving over the previous result. For \(m>1\), we present an algorithm that runs in \(O(\frac{n}{\varepsilon }\log (\frac{1}{\varepsilon })+m(\frac{1}{\varepsilon })^{O(\min (\sqrt{m},\frac{1}{\varepsilon }))})\) time.KeywordsMaximum coverageGeometric set coverPolynomial-time approximation scheme