We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of a fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3 2 is possible for strip packing in polynomial time (assuming P @math NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a @math -approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm. In this paper, we make further progress on the PPT approximability of strip packing, by presenting a @math -approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by @math , achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3 2 for the case with rotations as well.
Strip packing has also been well studied for higher dimensions. The present best asymptotic approximation for 3-D strip packing is due to Jansen and Pr " a del @cite_33 who presented a 1.5-approximation extending techniques from 2-D bin packing.
@cite_33
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We study the 3-dimensional Strip Packing problem: Given a list of n boxes b 1,…,b n of the width w i ≤ 1, depth d i ≤ 1 and an arbitrary length l i . The objective is to pack all boxes into a strip of the width and depth 1 and infinite length, so that the packing length is minimized. The boxes may not overlap or be rotated. We present an improvement of the current best asymptotic approximation ratio of 1.692 by [2] with an asymptotic 3 2 + e-approximation for any e > 0.