Two for One: Tight Approximation of 2D Bin Packing

Jansen, Klaus; Prädel, Lars; Schwarz, Ulrich M.

doi:10.1007/978-3-642-03367-4_35

Cited by 13 publications

(7 citation statements)

References 18 publications

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“…Harren & van Stee [6] also developed a non-asymptotic 2-approximation with rotations. Independently this approximation guarantee is also achieved for the version without rotations by Harren & van Stee [8] and Jansen et al [9]. These results match the non-asymptotic lower bound of this problem, unless P = N P.…”

Section: Introductionsupporting

confidence: 76%

New Approximability Results for Two-Dimensional Bin Packing

Jansen¹,

Prädel²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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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.

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Section: Introductionsupporting

confidence: 76%

New Approximability Results for Two-Dimensional Bin Packing

Jansen¹,

Prädel²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The weighted version does not admit an AFPTAS and an approximation algorithm by Jansen and Zhang [14] guarantees a ratio of 2+ε for every ε > 0. For the 2D bin packing, Jansen et al [12] gave a 2-approximation, and Bansal et al [3] designed a randomized algorithm with an asymptotic approximation ratio of about 1.525, improving the previous ratio 1.691 by Caprara [6,7]. However, 2D bin packing does not admit an AFPTAS [4,9].…”

Section: Introductionmentioning

confidence: 99%

Packing anchored rectangles

Dumitrescu

Tóth

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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Let S be a set of n points in the unit square [0, 1] 2 , one of which is the origin. We construct n pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in S, and the rectangles jointly cover at least a positive constant area (about 0.09). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2. IntroductionWe consider a rectangle packing problem popularized by Peter Winkler [18,19,20], which has been open for decades. It is a one-round game between Alice and Bob. First, Alice chooses a finite point set S in the unit square U = [0, 1] 2 in the plane, including the origin, that is, (0, 0) ∈ S ( Fig. 1(a)). Then Bob chooses an axisparallel rectangle r(s) ⊆ U for each point s ∈ S such that s is the lower left corner of r(s), and the interior of r(s) is disjoint from all other rectangles ( Fig. 1(b)). The rectangle r(s) is said to be anchored at s, but r(s) contains no point from S in its interior. It is conjectured that for any finite set S ⊂ U , (0, 0) ∈ S, Bob can choose such rectangles that jointly cover at least half of U [1,18,19,20]. However, it has not even been known whether Bob can always cover at least a positive constant area. It is clear that Bob cannot always cover

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“…The algorithm of [66] finds a feasible solution within 2N * bins in polynomial time by performing a separation between large and small items based on their area; it exploits as subroutines the Steinberg's algorithm [120] and the Next Fit Decreasing Height (NFDH) [37] procedure designed for strip packing, whereas the algorithm of [13] is used to guarantee the approximation ratio in the asymptotic case. 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.…”

Section: Single Solution Approximationmentioning

confidence: 99%

“…As noticed in [12] (Proposition 2.1), this problem can be approximated by heuristics for the traditional 2BP problem. In fact, 2BP can be 1-approximated both in the oriented [71] and the non-oriented [66] case. The algorithm of [66] finds a feasible solution within 2N * bins in polynomial time by performing a separation between large and small items based on their area; it exploits as subroutines the Steinberg's algorithm [120] and the Next Fit Decreasing Height (NFDH) [37] procedure designed for strip packing, whereas the algorithm of [13] is used to guarantee the approximation ratio in the asymptotic case.…”

Section: Single Solution Approximationmentioning

confidence: 99%