Online Packing to Minimize Area or Perimeter
Mikkel Abrahamsen, Lorenzo Beretta

TL;DR
This paper studies online algorithms for packing axis-parallel rectangles to minimize bounding box area or perimeter, providing bounds, algorithms, and analysis for various cases including rotations, aspect ratios, and minimum edge lengths.
Contribution
It introduces new bounds and algorithms for online rectangle packing, especially for minimizing area, and improves analysis for special cases like squares and bounded aspect ratios.
Findings
Perimeter minimization algorithms with competitive ratio just under 4.
Lower bounds showing asymptotic ratios of at least 8(1) for area minimization.
Algorithms matching these bounds for various rectangle constraints.
Abstract
We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90 degrees or translations only. For the perimeter version we give algorithms with an absolute competitive ratio slightly less than 4 when only translations are allowed and when rotations are also allowed. We then turn our attention to minimizing the area and show that the asymptotic competitive ratio of any algorithm is at least , where is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases. We also show that…
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