Packing, Hitting, and Coloring Squares
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Given a family of squares in the plane, their packing problem asks for the maximum number, ν, of pairwise disjoint squares among them, while their hitting problem asks for the minimum number, τ, of points hitting all of them, τ≥ν. Both problems are NP-hard even if all the rectangles are unit squares and their sides are parallel to the axes. The main results of this work are providing the first bounds for the τ / ν ratio on not necessarily axis-parallel squares. We establish an upper bound of 6 for unit squares and 10 for squares of varying sizes. The worst ratios we can provide with examples are 3 and 4, respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval [3/2,2] for unit squares and [3/2,4] for arbitrary squares. The new bounds necessitate a mixture of novel and classical techniques of possibly extendable use. Furthermore, we study rectangles with a bounded "aspect ratio", where the aspect ratio of a rectangle is the larger side of a rectangle divided by its smaller side. We improve on the well-known best τ/ν bound, which is quadratic in terms of the aspect ratio. We reduce it from quadratic to linear for rectangles, even if they are not axis-parallel, and from linear to logarithmic, for axis-parallel rectangles. Finally, we prove similar bounds for the chromatic numbers of squares and rectangles with a bounded aspect ratio.
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