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- Creator:
- Balas, Kevin
- Description:
- In this thesis, we deal with problems involving finding the maximum area covered when packing rectangles into a bounding box, each containing a specified representative point. Given a set of n points in the unit square, U = [0, 1]^2, we choose n interior-disjoint axis parallel rectangles each containing one point and seek the packing with maximal area. There are several variants of the problem, depending on the constraints put upon the rectangles. For example, arbitrary rectangles, anchored rectangles (where the representative point is a vertex) or squares. We look at how to generate all maximal anchored rectangle packings for a given point set and show that we need only compare those packings in which all vertices, from all rectangles, lie on grid points induced by the given point set. Our main result is an exponential upper bound for the number of lower left anchored maximal rectangle packings for a given point set in the unit square. We revisit some previous results using squares instead of rectangles and introduce upper and lower bounds for the area of anchored and unanchored square packings.
- Resource Type:
- Thesis
- Campus Tesim:
- Northridge
- Department:
- Mathematics

- Creator:
- Dumitrescu, Adrian, Balas, Kevin, and Toth, Csaba D.
- Description:
- 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.
- Resource Type:
- Article
- Identifier:
- 1572-5286
- Campus Tesim:
- Northridge

- Creator:
- Dumitrescu, Adrian, Balas, Kevin, and Toth, Csaba D.
- Description:
- 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 squarepacking 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.
- Resource Type:
- Article
- Identifier:
- 1572-5286
- Campus Tesim:
- Northridge