New Approximability Results for Two-Dimensional Bin Packing
We study the two-dimensional bin packing problem: Given a list of n rectangles the objective is to find a feasible, i.e. axis-parallel and non-overlapping, packing of all rectangles into the minimum number of unit sized squares, also called bins. Our problem consists of two versions; in the first version it is not allowed to rotate the rectangles while in the other it is allowed to rotate the rectangles by 90∘, i.e. to exchange the widths and the heights. Two-dimensional bin packing is a generalization of its one-dimensional counterpart and is therefore strongly NP-hard. Furthermore Bansal et al. showed that even an APTAS is ruled out for this problem, unless P=NP. This lower bound of asymptotic approximability was improved by Chlebik and Chlebikova to values 1+1/3792 and 1+1/2196 for the version with and without rotations, respectively. On the positive side there is an asymptotic 1.69.. approximation by Caprara without rotations and an asymptotic 1.52... approximation by Bansal et al.for both versions. We give a new asymptotic upper bound for both versions of our problem: For any fixed ε and any instance that fits optimally into OPT bins, our algorithm computes a packing into (3/2+ε)⋅OPT+69 bins in the version without rotations and (3/2+ε)⋅OPT+39 bins in the version with rotations. The algorithm has polynomial running time in the input length. In our new technique we consider an optimal packing of the rectangles into the bins. We cut a small vertical or horizontal strip out of each bin and move the intersecting rectangles into additional bins. This enables us to either round the widths of all wide rectangles, or the heights of all long rectangles in this bin. After this step we round the other unrounded side of these rectangles and we achieve a solution with a simple structure and only few types of rectangles. Our algorithm initially rounds the instance and computes a solution that nearly matches the modified optimal solution.