On Greedily Packing Anchored Rectangles

TL;DR

This paper improves the analysis of a greedy algorithm for packing anchored rectangles in a unit square, increasing guaranteed coverage from 9.1% to 39%, but also shows instances where coverage is below 43.3%, indicating limits to the approach.

Contribution

The authors provide a significantly improved analysis of a greedy algorithm for anchored rectangle packing, raising the guaranteed coverage and identifying the algorithm's limitations.

Findings

Algorithm guarantees at least 39% coverage of the unit square.

Constructed instances where coverage drops below 43.3%.

Analysis of tile density informs potential for related algorithms.

Abstract

Consider a set P of points in the unit square U, one of them being the origin. For each point p in P you may draw a rectangle in U with its lower-left corner in p. What is the maximum area such rectangles can cover without overlapping each other? Freedman [1969] posed this problem in 1969, asking whether one can always cover at least 50% of U. Over 40 years later, Dumitrescu and T\'oth [2011] achieved the first constant coverage of 9.1%; since then, no significant progress was made. While 9.1% might seem low, the authors could not find any instance where their algorithm covers less than 50%, nourishing the hope to eventually prove a 50% bound. While we indeed significantly raise the algorithm's coverage to 39%, we extinguish the hope of reaching 50% by giving points for which the coverage is below 43.3%. Our analysis studies the algorithm's average and worst-case density of so-called tiles, which represent the area where a given point can freely choose its maximum-area rectangle. Our approachis comparatively general and may potentially help in analyzing related algorithms.