On Greedily Packing Anchored Rectangles

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

Packing boundary-anchored rectangles and squares

Consider a set P of n points on the boundary of an axis-aligned square Q. We study the boundary-anchored packing problem on P in which the goal is to find a set of interior-disjoint axis-aligned rectangles in Q such that each rectangle is anchored (has a corner at some point in P), each point in P is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point p in P if one of its corners coincides with p. In this paper, we show how to solve this problem in time linear in n, provided that the points of P are given in sorted order along the boundary of Q. We also consider the problem for anchoring squares and give an O(n4)-time algorithm when the points in P lie on two opposite sides of Q