### Improved Approximation Algorithm for Two-Dimensional Bin Packing [chapter]

Nikhil Bansal, Arindam Khan
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
We study the two-dimensional bin packing problem with and without rotations. Here we are given a set of two-dimensional rectangular items I and the goal is to pack these into a minimum number of unit square bins. We consider the orthogonal packing case where the edges of the items must be aligned parallel to the edges of the bin. Our main result is a 1.405approximation for two-dimensional bin packing with and without rotation, which improves upon a recent 1.5 approximation due to Jansen and
more » ... el. We also show that a wide class of rounding based algorithms cannot improve upon the factor of 1.5. Related previous work. In their celebrated work, de la Vega and Lueker [11] gave the first APTAS for the 1-D bin packing problem. This was substantially improved by Karmarkar and Karp [20] who gave a guarantee of Opt + O(log 2 Opt). Very recently, this has been improved by Rothvoss [26] to Opt + O(log Opt). On the other hand, the possibility of an algorithm with an Opt + 1 guarantee is still open. The 2-D case is substantially different from the 1-D case. Bansal et al. [2] showed that no APTAS is possible unless P=NP. On the positive side, there has also been a long sequence of works giving improved algorithms. Until the mid 90's the best known bound was a 2.125 approximation [9] , which was improved by Kenyon and Rémila [22] to a 2 + approximation for any > 0. An important breakthrough was achieved by Caprara [5], who gave an algorithm that achieves an asymptotic approximation ratio of T ∞ + ≈ 1.69103 + . Here T ∞ is the well-known "Harmonic" constant that appears ubiquitously in the context of bin packing. This was later improved by Bansal et al. [3] to (ln T ∞ +1) ≈ 1.52 by combining the algorithm of Caprara [5] with a general approximation method for set-covering problems known as Round-and-Approx, that we will also consider in this paper. Recently Jansen and Prädel [16] improved this guarantee further to give a 1.5-approximation algorithm. Their algorithm is based on exploiting several non-trivial structural properties of how items can be packed in a bin. This is the best algorithm known so far, and holds both for the case with and without rotations. We remark that there is still a huge gap between these upper bounds and known lower bounds. In particular, the best known explicit lower bound on the asymptotic approximation for 2-D BP is currently 1+1/3792 and 1+1/2196 for the versions with and without rotations respectively [7]. Our Results. Our main result is an improved algorithm for the 2-D bin packing problem. In particular we show the following. Theorem 1.1. There is a polynomial time algorithm with an asymptotic approximation ratio of ln(1.5) + 1 ≈ 1.405 for 2-D bin packing. This holds both for the version with and without rotations.