Anchored Rectangle And Square Packings

For points in the unit square , an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles such that point is a corner of the rectangle (that is, is anchored at ) for . We show that for every set of points in , there is an anchored rectangle packing of area at least , and for every , there are point sets for which the area of every anchored rectangle packing is at most . The maximum area of an anchored square packing is always at least and sometimes at most . The above constructive lower bounds immediately yield constant-factor approximations, of for rectangles and for squares, for computing anchored packings of maximum area in time. We prove that a simple greedy strategy achieves a -approximation for anchored square packings, and for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS for anchored rectangle packings in time and a PTAS for anchored square packings in time.