Aligning Two Convex Figures to Minimize Area or Perimeter

Abstract

Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕP ∪ Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact nearlinear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1 + ε)-approximation in time O(ε −1/2 log n + ε −3/2 log a (1/ε)) if the two sets are convex polygons with n vertices in total, where a ∈ {0, 1, 2} depending on the version of the problem.

Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

“…To our knowledge, most of the problems we consider have not been studied before by any other researchers. 1 Problem (P2) was studied recently for the special case K = 2 [11]; we consider the general case of arbitrary K (Section 3). On the contrary, for problem (P6), algorithms for general K are known [19,47]; we improve their running times for the special case K = 2 (Section 4.2).…”

“…Convex suitcase Finding a minimum convex enclosure for stacked items (our problem (P2)) was studied by Ahn and Cheong [11] for the special case of two items (K = 2). They gave exact and approximation algorithms for various versions of the problem; for our problem (P2) (minimizing the area of the convex hull of the union, while allowing overlap), [11] gives a PTAS.…”

“…They gave exact and approximation algorithms for various versions of the problem; for our problem (P2) (minimizing the area of the convex hull of the union, while allowing overlap), [11] gives a PTAS. In Section 3 we show that for general K the problem is NP-hard and give a PTAS for perimeter minimization.…”

“…It is shown in [11] (see also [12,Theorem 5]) that the perimeter of the convex hull of the union of rectangles is a convex function with respect to translation of the items. Thus, in an optimal solution the items' centers should be at the same point, and the problem reduces to finding optimal rotations for the items.…”

Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons

“…So, only for a constant number of objects polynomial-time algorithms are known, see [1,2,14,3,8,9,10,7]. Therefore, numerous approximation algorithms and heuristics have been investigated, mostly in the operations research and combinatorial optimization communities and mostly on axis-parallel rectangles under translation for a given container or strip.…”