Packing anchored rectangles

Abstract

Let S be a set of n points in the unit square [0, 1] 2 , one of which is the origin. We construct n pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in S, and the rectangles jointly cover at least a positive constant area (about 0.09). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2. IntroductionWe consider a rectangle packing problem popularized by Peter Winkler [18,19,20], which has been open for decades. It is a one-round game between Alice and Bob. First, Alice chooses a finite point set S in the unit square U = [0, 1] 2 in the plane, including the origin, that is, (0, 0) ∈ S ( Fig. 1(a)). Then Bob chooses an axisparallel rectangle r(s) ⊆ U for each point s ∈ S such that s is the lower left corner of r(s), and the interior of r(s) is disjoint from all other rectangles ( Fig. 1(b)). The rectangle r(s) is said to be anchored at s, but r(s) contains no point from S in its interior. It is conjectured that for any finite set S ⊂ U , (0, 0) ∈ S, Bob can choose such rectangles that jointly cover at least half of U [1,18,19,20]. However, it has not even been known whether Bob can always cover at least a positive constant area. It is clear that Bob cannot always cover

A better lower bound for Lower-Left Anchored Rectangle Packing

“…Our analysis improves on that of Dumitrescu and Tóth [11] in three ways. We introduce a variable α that varies the definition of the tips of β-tiles (to be defined later), we bound the area of such tiles by parallelograms instead of trapezoids, and we bound all β-tiles at once with the use of the inequality of arithmetic and geometric means.…”

“…In 1969, Allen Freedman [17] conjectured that no matter what set of points Alice chooses, Bob can always find a set of rectangles that satisfies the conditions and covers an area of at least half of the unit square. This conjecture remains open and it was even only in 2012 that Dumitrescu and Tóth [10,11] published the first result that shows that Bob can always cover any constant size area at all. Since then, the question of packing anchored rectangles in the unit-square has attracted more attention, yet, so far, no one has improved the lower bound on what Bob can achieve.…”