Anchored rectangle and square packings

Abstract

For points p 1 , . . . , p n in the unit square [0,1] 2 , an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r 1 , . . . , r n ⊆ [0, 1] 2 such that point p i is a corner of the rectangle r i (that is, r i is anchored at p i ) for i = 1, . . . , n. We show that for every set of n points in [0,1] 2 , there is an anchored rectangle packing of area at least 7/12 − O(1/n), and for every n ∈ N, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 − ε for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n O(1/ε) and exp(poly(log(n/ε))) time, respectively.

Packing boundary-anchored rectangles and squares

“…An outline. It is easy to see [1] that in any rectangle packing the boundaries of rectangles must lie on the grid Γ: the union of the boundary of Q and the extension of inward rays from all points until they hit the opposite boundary. For each point p ∈ P , there are O(n 2 ) potential rectangles of Γ anchored at p, and so we have a total of O(n 3 ) candidate rectangles from which we must pick an independent set (with respect to their intersection graph) such that the sum of the weights (defined to be the area of each rectangle) is maximized.…”

“…If S covers all of Q, then by Lemma 3.7 one of (i)-(iii) holds. If (i) holds, then the corner in P will be chosen under rule (1). (In these and all other cases, "chosen" means "after a suitable rotation and/or reflection".)…”

“…It is not known whether this problem is NP-hard. The best known approximation algorithm for this problem achieves ratio 7/12 − ε due to Balas et al [1], who also studied several variants of this problem.…”

“…Rectangle packing problems have applications in map labeling [7,10]. Balas et al [1] studied several variants of the anchored packing problem; namely, the lower-left anchored rectangle packing problem in which points of P are required to be on the lower-left corners of the rectangles in R, the anchored square packing problem in which every anchored rectangles is required to be a square, and the lower-left anchored square packing problem which is a combination the two previous problems. For the lower-left rectangle packing problem, Freedman [9] conjectured that there is a solution that covers 50% of the area of Q.…”

“…The best known lower bound of 9.1% of the area of Q is due to Dumitrescu and Tóth [4]. Balas et al [1] presented approximation algorithms with ratios (7/12 − ε) and 5/32 for anchored rectangles and anchored square, respectively. They also presented a 1/3-approximation algorithm for the lower-left anchored square packing problem, and proved that the analysis of the approximation factor is tight.…”