On the area of overlap of translated polygons
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We present an efficient solution method for packing d-dimensional polytopes within the bounds of a polytope container. The central geometric operation of the method is an exact one-dimensional translation of a given polytope to a position which minimizes its volume of overlap with all other polytopes. We give a detailed description and a proof of a simple algorithm for this operation in which one only needs to know the set of (d-1)-dimensional facets in each polytope. Handling non-convex polytopes or even interior holes is a natural part of this algorithm. The translation algorithm is used as part of a local search heuristic and a meta-heuristic technique, guided local search, is used to escape local minima. Additional details are given for the three-dimensional case and results are reported for the problem of packing polyhedra in a rectangular parallelepiped. Utilization of container space is improved by an average of more than 14 percentage points compared to previous methods. The translation algorithm can also be used to solve the problem of maximizing the volume of intersection of two polytopes given a fixed translation direction. For two polytopes with complexity O(n) and O(m) and a fixed dimension, the running time is O(nmlog(nm)) for both the minimization and maximization variants of the translation algorithm.