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Maximum Overlap of Convex Polytopes under Translation
"... We study the problem of maximizing the overlap of two convex polytopes under translation in R d for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε>0, finds an overlap at least the optimum minus ε and reports the translati ..."
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We study the problem of maximizing the overlap of two convex polytopes under translation in R d for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε>0, finds an overlap at least the optimum minus ε and reports the translation realizing it. The running time is O(n ⌊d/2⌋+1 log d n) with probability at least 1 − n −O(1),whichcanbeimprovedtoO(n log 3.5 n)inR 3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the bigO constants in these bounds are independent of ε.
Scandinavian Thins on Top of Cake: New and improved algorithms for stacking and packing
"... We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest conve ..."
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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NPhard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.