Distant Representatives for Rectangles in the Plane

The input to the distant representatives problem is a set of \(n\) objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are axis-aligned rectangles, we give polynomial time constant-factor approximation algorithms for the \(L_1\), \(L_2\), and \(L_\infty\) distance measures. We also prove lower bounds on the approximation factors that can be achieved in polynomial time (unless P = NP).