New Approximability Results for Two-Dimensional Bin Packing

Abstract

We study the two-dimensional bin packing problem: Given a list of n rectangles the objective is to find a feasible, i.e. axisparallel 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 N Phard. Furthermore Bansal et al. [2] showed that even an APTAS is ruled out for this problem, unless P = N P. This lower bound of asymptotic approximability was improved by Chlebík & Chlebíková 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 (1.5 + ε) · OPT + 69 bins with 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.

Pricing-based primal and dual bounds for selected packing problems

“…The same absolute factor for the oriented case is attained by the algorithm of [71], which strongly relies on the PTAS for 2D knapsack problem presented in [14] and the techniques for rectangle packing described in [72]. The area and the sizes of the items are concurrently combined to identify a more extensive separation, while the Steinberg's and NFDH algorithms are exploited again and the asymptotic approximation ratio is ensured through the algorithm of [70] for large optimal values.…”

Online Square-into-Square Packing

“…Arguably, bin packing is the twodimensional packing problem that has received the most attention from an algorithmic perspective. See [10,9,15,13,8,5,3,11,4,47,31,28,7] for particularly relevant work. Most of these papers consider offline problems, with notable exceptions already cited above.…”

A hybrid feasibility constraints-guided search to the two-dimensional bin packing problem with due dates

“…Bin packing (BP) is a classical strongly N P-hard combinatorial optimization problem (Jansen & Pradel, 2016;Johnson et al, 1974). It consists in packing a set of items into as few bins as possible.…”