Bundling Three Convex Polygons to Minimize Area or Perimeter

Abstract

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 τ ₀,…,τ _k − 1 of P ₀,…,P _k − 1 such that the translated copies τ _i P _i 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 P ₁ and P ₂ can be decomposed into O(n ²) 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(n ²)-time algorithm that returns an optimal pair of translations minimizing the area or the perimeter of the corresponding convex hull.

The research by H.-K. Ahn and D. Park was supported by NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea.