A Harmonic Algorithm for the 3D Strip Packing Problem

In the three dimensional strip packing problem, we are given a set of three dimensional rectangular items and a three dimensional box B. The goal is to pack all the items in B such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of B and cannot be rotated. Building upon Caprara’s work [5, 6] for the two dimensional bin packing problem, we obtain an algorithm that given any > 0, achieves an approximation of T∞ + ≈ 1.69103 + , where T∞ is the well known number that occurs naturally in the context of bin packing. The previously known algorithms for this problem had worst case performance guarantees of 2 [12], 2.67 [19], 2.89 [16] and 3.25 [15]. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance I and the strip packing solutions for the corresponding instance obtained from I by applying the Harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the T∞ + approximation for two dimensional bin packing due to Caprara [5, 6]. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for two dimensional strip packing due to Kenyon and Rémila [14].