Translational polygon containment and minimal enclosure using linear programming based restriction
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Packing Convex Polygons into Rectangular Boxes
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Stacking and bundling two convex polygons
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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Given a set ${\mathcal P} =\{P_0,\ldots,P_{k-1}\}$ of k convex polygons having n vertices in total in the plane, we consider the problem of finding k translations τ0,…,τk−1 of P0,…,Pk−1 such that the translated copies τiPi are pairwise disjoint and the area or the perimeter of the convex hull of $\bigcup_{i=0}^{k-1}\tau_iP_i$ is minimized. When k=2, the problem can be solved in linear time but no previous work is known for larger k except a hardness result: it is NP-hard if k is part of input. We show that for k=3 the translation space of P1 and P2 can be decomposed into O(n2) cells in each of which the combinatorial structure of the convex hull remains the same and the area or perimeter function can be fully described with O(1) complexity. Based on this decomposition, we present a first O(n2)-time algorithm that returns an optimal pair of translations minimizing the area or the perimeter of the corresponding convex hull.