On Greedily Packing Anchored Rectangles
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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 and Tóth [2011] achieved the first constant coverage of 9.1 significant progress was made. While 9.1 find any instance where their algorithm covers less than 50 hope to eventually prove a 50 algorithm's coverage to 39 points for which the coverage is below 43.3 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.
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