Distant Representatives for Rectangles in the Plane

TL;DR

This paper presents polynomial-time approximation algorithms for selecting representative points in a set of rectangles in the plane, maximizing the minimum pairwise distance under various metrics, and establishes related computational hardness bounds.

Contribution

It introduces new polynomial-time algorithms for the distant representatives problem specifically for rectangles and provides lower bounds on approximation factors under standard complexity assumptions.

Findings

Polynomial-time constant-factor approximation algorithms for rectangles.

Lower bounds on approximation factors unless P=NP.

Results applicable to L1, L2, and L_infinity distances.

Abstract

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).