On Greedily Packing Anchored Rectangles
Christoph Damerius, Dominik Kaaser, Peter Kling, Florian Schneider

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…
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