On Greedily Packing Anchored Rectangles

Abstract

Consider a set P of points in the unit square U = [1, 0), one of them being the origin. For each point p ∈ P you may draw an axis-aligned rectangle in U with its lower-left corner being p. What is the maximum area such rectangles can cover without overlapping each other? Freedman [18] posed this problem in 1969, asking whether one can always cover at least 50% of U . Over 40 years later, Dumitrescu and Tóth [12] 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 its coverage stays below 43.3%. Our analysis studies the algorithm’s average and worst-case density of so-called tiles, which represent the staircase polygons in which a point can freely choose its maximum-area rectangle. Our approach is comparatively general and may potentially help in analyzing related algorithms. 2012 ACM Subject Classification Theory of computation → Computational geometry