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. 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 N P-hard. 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 [5] 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 [4] without rotations and an asymptotic 1.52... approximation by Bansal et al. [1] 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.

Improved Approximation Algorithm for Two-Dimensional Bin Packing

“…However, this does not rule out the possibility of an Opt + 1 guarantee, and hence it is insightful to consider the asymptotic approximation ratio (AAR, denoted by R The main idea behind theorem 1.1 is to show that the round and approx framework introduced by [3] (we describe this in section 2) can be applied to the result of Jansen and Prädel [16]. Roughly speaking, this framework states that given a packing problem, if (i) the configuration LP for the problem (with the original item sizes) can be solved up to error 1 + for any > 0, and (ii) there is a ρ approximation for the problem that is subset-oblivious; then one can obtain a (1 + ln ρ) asymptotic approximation for the problem.…”

“…However, the notion of subsetobliviousness as defined in [3] is based on various properties of dual-weighting functions, making it somewhat tedious to apply and also limited in scope (e.g. it is unclear to us how to apply this method directly to the algorithm of [16]). …”

“…Rounding of items to O(1) types has been used implicitly [4] or explicitly [11,20,5,16,21], in almost all bin packing algorithms. There are typically two types of rounding: either the size of an item in some coordinate (such as width or height) is rounded up in an instanceoblivious way (e.g.…”

Two for One: Tight Approximation of 2d Bin Packing

“…Clearly, the total width of items at least as high as r is larger than 1−w j , otherwise, the overlap would not have occurred. By Lemma 12, setting γ = 1 − h < 1/2, the total height of wide non-high items of width at least w j is then at most 2(1 − h ), however, it is also at least (12) which contradicts the assumption of overlap.…”

“…As mentioned above, Jansen and Prädel have given in [12] an algorithm that for any and any instance obtains in polynomial time a solution with at most (1.5 + )OPT + 69 bins. Let us consider the case that OPT is at least the threshold value k := 140.…”