A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability

Bansal, Nikhil; Caprara, Alberto; Jansen, Klaus; Prädel, Lars; Sviridenko, Maxim

doi:10.1007/978-3-642-10631-6_10

Cited by 25 publications

(44 citation statements)

References 31 publications

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“…The PTAS in [13] for geometric knapsack for squares (and many other algorithms for geometric problems, e.g., [2,16,3]) applies the mentioned shifting technique to distinguish squares into large and small squares. Eventually, it is shown that there is a (1 + )-approximative packing which is structured into O (1) large squares and O (1) cells.…”

Section: Our Techniquesmentioning

confidence: 99%

“…However, there are PTASs for the resource augmentation setting in both [9] or in only one [12] dimension, and for the case that the profit of each item equals its area [3]. The running time of all these (1 + )-approximation algorithms is Ω(n 1/ 1/ ).…”

Section: Other Related Workmentioning

confidence: 99%

“…[10]. Both of these results use the doublyexponential algorithm in [3] as a subroutine. Moreover, for strip packing there is an asymptotic FPTAS [17] and a (1.4 + )-approximation algorithm in (doublyexponential) pseudo-polynomial time.…”

Section: Other Related Workmentioning

confidence: 99%

“…Guessing the number of cells (block cells and elongated cells) takes time O(1/ 10 F ( ) 3 ), guessing the number of frames for each elongated cell takes total time (1/ ) O(1/ 10 F ( ) 3 ) , and guessing the number of subframes for each frame takes total time (1/ ) O(1/ 14 F ( ) 3 ) . This gives a total running time of…”

Section: Computing the Packingmentioning

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: Our Techniquesmentioning

confidence: 99%

Section: Other Related Workmentioning

confidence: 99%

Section: Other Related Workmentioning

confidence: 99%

Section: Computing the Packingmentioning

confidence: 99%

See 2 more Smart Citations

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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“…In recent years, there has been increasing interest in extensions of packing problems such as strip packing [1,2,14,17,19], knapsack [3,15,18] and bin packing [4,5,6,9,20], to multiple independent criteria (vector packing) or multiple dimensions (geometric packing).…”

Section: Introductionmentioning

confidence: 99%

Two for One: Tight Approximation of 2d Bin Packing

Harren

Jansen

Prädel

et al. 2013

Int. J. Found. Comput. Sci.

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In this paper, we study the two-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of unit bins without rotating the rectangles. Beyond its theoretical appeal, this problem has many practical applications, for example in print layout and VLSI chip design.We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the problem where rotations are allowed and also matches the known lower bound of approximability. Our approach makes strong use of a PTAS for a related 2D knapsack problem and a new algorithm that can pack instances into two bins if OPT = 1.

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