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Rectangle packing problems with distinct representatives
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.