Improved Approximation Algorithm for Two-Dimensional Bin Packing

Bansal, Nikhil; Khan, Arindam

doi:10.1137/1.9781611973402.2

Cited by 47 publications

(57 citation statements)

References 15 publications

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“…Roughly speaking, this works because the remaining items are packed using the APTAS which is rounding based. The proof has some additional technical difficulties compared to the previous result on 2-dimensional geometric bin packing [3], due to skewed items that are large in one dimension and small in another. This is described in Section 6.1.…”

Section: Overview and Roadmapmentioning

confidence: 83%

“…In the original approach of [2], the definition of a suitable algorithm involved a certain technical notion of subset-obliviousness. Later, Bansal and Khan [3] simplified and extended the applicability of this approach (in the context of geometric bin-packing) by showing that any rounding based algorithm is subset-oblivious. As rounding-based algorithms cannot beat d and extending R&A beyond such algorithms was unclear, this was a major barrier to improving the ratio of 1 + ln(d).…”

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confidence: 99%

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Improved Approximation for Vector Bin Packing

Bansal¹,

Eliáš²,

Khan³

2015

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

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We study the d-dimensional vector bin packing problem, a well-studied generalization of bin packing arising in resource allocation and scheduling problems. Here we are given a set of d-dimensional vectors v1, . . . , vn in [0,1] d , and the goal is to pack them into the least number of bins so that for each bin B, the sum of the vectors in it is at most 1 in every dimension, i.e., || v i ∈B vi||∞ ≤ 1. For the 2-dimensional case we give an asymptotic approximation guarantee of 1 + ln(1.5) + ǫ ≈ (1.405 + ǫ), improving upon the previous bound of 1 + ln 2 + ǫ ≈ (1.693 + ǫ). We also give an almost tight (1.5 + ǫ) absolute approximation guarantee, improving upon the previous bound of 2 [23]. For the d-dimensional case, we get a 1.5 + ln() + ǫ ≈ 0.807 + ln(d + 1) + ǫ guarantee, improving upon the previous (1+ln d+ǫ) guarantee [2]. Here (1 +ln d) was a natural barrier as rounding-based algorithms can not achieve better than d approximation. We get around this by exploiting various structural properties of (near)-optimal packings, and using multi-objective multi-budget matching based techniques and expanding the Round & Approx framework to go beyond rounding-based algorithms. Along the way we also prove several results that could be of independent interest.

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Section: Overview and Roadmapmentioning

confidence: 83%

mentioning

confidence: 99%

Improved Approximation for Vector Bin Packing

Bansal¹,

Eliáš²,

Khan³

2015

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

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“…We can thus again use our online algorithm A 2-D to obtain the following adaption of Theorem 17. The approximation algorithm of Bansal and Khan [4] used above can also handle the case of rotation and thus is an 1.405-approximation for 2-D Bin Packing With Rotations. We can thus use Theorem 7 with γ = 1.405 and β = 48 /5 to conclude the following theorem.…”

Section: Rotationsmentioning

confidence: 99%

Robust Online Algorithms for Certain Dynamic Packing Problems

Berndt

Dreismann

Grage

et al. 2020

Approximation and Online Algorithms

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Online algorithms that allow a small amount of migration or recourse have been intensively studied in the last years. They are essential in the design of competitive algorithms for dynamic problems, where objects can also depart from the instance. In this work, we give a general framework to obtain so called robust online algorithms for these dynamic problems: these online algorithm achieve an asymptotic competitive ratio of γ + ǫ with migration O(1/ǫ), where γ is the best known offline asymptotic approximation ratio. In order to use our framework, one only needs to construct a suitable online algorithm for the static online case, where items never depart. To show the usefulness of our approach, we improve upon the best known robust algorithms for the dynamic versions of generalizations of Strip Packing and Bin Packing, including the first robust algorithm for general d-dimensional Bin Packing and Vector Packing. ACM Subject Classification Theory of computation → Online algorithms

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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]. Another related problem is strip packing, where squares need to be packed into a strip of width 1 such that the height of the packing is minimized.…”

Section: Other Related Workmentioning

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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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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Improved Approximation Algorithm for Two-Dimensional Bin Packing

Cited by 47 publications

References 15 publications

Improved Approximation for Vector Bin Packing

Improved Approximation for Vector Bin Packing

Robust Online Algorithms for Certain Dynamic Packing Problems

Faster approximation schemes for the two-dimensional knapsack problem

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