Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes

Bansal, Nikhil; Correa, José R.; Kenyon, Claire; Sviridenko, Maxim

doi:10.1287/moor.1050.0168

Cited by 104 publications

(140 citation statements)

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“…For the multi-dimensional case of packing hypercubes, there is an APTAS [4], and it is also shown that there exists an algorithm that packs (two-dimensional) rectangles into the optimal number of bins using resource augmentation. For the case of rectangles the currently best known result is a (1.405 + )-approximation [5].…”

Section: Other Related Workmentioning

confidence: 99%

“…There are many well-studied geometric problems in combinatorial optimization, for instance the natural geometric generalizations of fundamental one-dimensional problems like Knapsack or Bin Packing. For many settings of those, there are polynomial time algorithms known that give an approximation guarantee of 1 + , e.g., [13,4,9]. A common theme in these PTASs is the shifting technique: items that need to be packed are classified into large and small squares such that intuitively the large squares are much larger than the small ones.…”

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confidence: 99%

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Faster approximation schemes for the two-dimensional knapsack problem

Heydrich¹,

Wiese²

2017

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

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An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + )-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1/ , i.e., Ω(n 2 2 1/ ). A double or triple exponential dependence on 1/ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ )-approximation algorithm with a running time of O (1) · n O(1) . In particular, the exponent of n in the running time does not depend on at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(2 2 1/ ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n 2 2 1/ ) and we improve both to O (1) · n O(1) . We complete our result by giving an algorithm for twodimensional knapsack for rectangles under (1 + )-resource augmentation. In this setting, we also improve the best known running time of Ω(n 1/ 1/ ) to O (1) · n O(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well.

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Section: Other Related Workmentioning

confidence: 99%

mentioning

confidence: 99%

Faster approximation schemes for the two-dimensional knapsack problem

Heydrich¹,

Wiese²

2017

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

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“…Bansal et al [2] devised a randomized algorithm with an asymptotic performance ratio of at most 1.525. As for the offline lower bound of the approximation ratio, Bansal et al [1] showed that the two-dimensional bin packing problem does not admit any asymptotic polynomial time approximation scheme.…”

Section: Related Workmentioning

confidence: 99%

Improved Online Algorithms for 1-Space Bounded 2-Dimensional Bin Packing

Zhang

Chen

Chin

et al. 2010

Algorithms and Computation

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Abstract. In this paper, we study 1-space bounded 2-dimensional bin packing and square packing. A sequence of rectangular items (square items, respectively) arrive over time, which must be packed into square bins of size 1×1. 90 • -rotation of an item is allowed. When an item arrives, we must pack it into an active bin immediately without any knowledge of the future items. The objective is to minimize the total number of bins used for packing all the items in the sequence. In the 1-space bounded variant, there is only one active bin for packing the current item. If the active bin does not have enough space to pack the item, it must be closed and a new active bin is opened. Our contributions are as follows: For 1-space bounded 2-dimensional bin packing, we propose an online packing strategy with competitive ratio 5.155, surpassing the previous 8.84-competitive bound. The lower bound of competitive ratio is also improved from 2.5 to 3. Furthermore, we study 1-space bounded square packing, which is a special case of the bin packing problem. We give a 4.5-competitive packing algorithm, and prove that the lower bound of competitive ratio is at least 8/3.

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“…These are the currently best-known approximation ratios for these problems. For packing squares into square bins, Bansal, Correa, Kenyon & Sviridenko [2] gave an asymptotic PTAS. On the other hand, the same paper showed the APXhardness of rectangle packing without rotations, thus no asymptotic PTAS exists unless P = N P. Chlebík & Chlebíková [4] were the first to give explicit lower bounds of 1 + 1/3792 and 1 + 1/2196 on the asymptotic approximability of rectangle packing with and without rotations, respectively.…”

Section: Introductionmentioning

confidence: 99%