Anchored rectangle and square packings

For points p1,…,pn in the unit square [0,1]2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r1,…,rn⊆[0,1]2 such that point pi is a corner of the rectangle ri (that is, ri is anchored at pi) 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(nlogn) 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 for anchored rectangle packings in exp(poly(ε−1logn)) time and a PTAS for anchored square packings in nO(1∕ε) time.