Stacking and Bundling Two Convex Polygons
Given two compact convex sets C1 and C2 in the plane, we consider the problem of finding a placement ϕC1 of C1 that minimizes the area of the convex hull of ϕC1 ∪ C2. We first consider the case where ϕC1 and C2 are allowed to intersect (as in “stacking” two flat objects in a convex box), and then add the restriction that their interior has to remain disjoint (as when “bundling” two convex objects together into a tight bundle). In both cases, we consider both the case where we are allowed to reorient C1, and where the orientation is fixed. In the case without reorientations, we achieve exact near-linear time algorithms, in the case with reorientations we compute a (1 + ε)-approximation in time O(ε− 1/2 log n + ε− 3/2 log ε− 1/2), if two sets are convex polygons with n vertices in total.
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