Packing Convex Polygons into Rectangular Boxes

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.…”

“…It remains to prove claim (3). To simplify the presentation we assume 0 < j < t; it is easily veried that a similar argument applies when j = 0 and when j = t. To get a bound on the area of Q j , let s p be the segment parallel to s(p j ) that splits Q j and passes through the point p where p j touches p j+1 .…”

Aligning Two Convex Figures to Minimize Area or Perimeter

“…(5). Using Lemma 1, we construct outer ε -approximations P and Q to P and Q in time O((T P + T Q )ε −1/2 ).…”

“…For packing k convex n-gons into a minimum area isothetic rectangle under translation, he gave an O(n k−1 log n) time algorithm using linear programming techniques. Later, Alt and Hurtado [5] considered the problem of packing convex polygons into a minimum size rectangle, where the size of a rectangle can either be its area or its perimeter. When overlap is allowed, they presented efficient algorithms whose running time is close to linear or O(n log n) even for an arbitrary number of polygons with n vertices in total.…”

Cutting a Country for Smallest Square Fit

“…Alt and Hurtado [1] discuss packing convex polygons into a rectangle of smallest size, measured either by area or by perimeter. They study both packing with overlap and without overlap.…”