Maximum Weight Convex Polytope
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We study the maximum weight convex polytope problem, in which the goal is to find a convex polytope maximizing the total weight of enclosed points. Prior to this work, the only known result for this problem was an O(n^3) algorithm for the case of 2 dimensions due to Bautista et al. We show that the problem becomes 𝒩𝒫-hard to solve exactly in 3 dimensions, and 𝒩𝒫-hard to approximate within n^1/2-ϵ for any ϵ > 0 in 4 or more dimensions. binary weights. We also give a new algorithm for 2 dimensions, albeit with the same O(n^3) running time complexity as that of the algorithm of Bautsita et al.
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