Bundling Three Convex Polygons to Minimize Area or Perimeter

  • Hee-Kap Ahn
  • Helmut Alt
  • Sang Won Bae
  • Dongwoo Park
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


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.


Convex Hull Feasible Region Convex Polygon Event Curve Combinatorial Structure 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Helmut Alt
    • 2
  • Sang Won Bae
    • 3
  • Dongwoo Park
    • 1
  1. 1.POSTECHSouth Korea
  2. 2.Freie Universität BerlinGermany
  3. 3.Kyonggi UniversitySouth Korea

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