There is no APTAS for 2-dimensional vector bin packing: Revisited

Ray, Arka Prabha

doi:10.48550/arxiv.2104.13362

Cited by 3 publications

(6 citation statements)

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“…Proof. The lemma is essentially proven by Ray [29] and follows from close inspection of his reduction. Indeed, the requests created in Ray's reduction are vectors of rational numbers whose common denominator D is at most 10((64n) 4 + 15) and whose numerators are also bounded by a polynomial of n. We multiply all requests a (i) by D, obtaining a (i) := Da (i) , and choose the bin capacity b := (D, D).…”

Section: Problem 35 (2d Vector Bin Packing (2d-vbp))mentioning

confidence: 88%

“…Ray [29,Theorem 6] proved that 2D-VBP does not have polynomial-time algorithms with asymptotic approximation ratio better than 600/599. In fact, he shows that the 600/599-gap version of 2D-VBP is NP-hard.…”

Section: Problem 35 (2d Vector Bin Packing (2d-vbp))mentioning

confidence: 99%

“…Unfortunately, even when all requests have identical start and end times and d = 2, the best known polynomial-time approximation algorithm for DVBP yields an asymptotic approximation factor of 1.405 + ε [3]. Unless P = NP, asymptotic (1 + ε)-approximations for arbitrarily small ε > 0 will require superpolynomial time [29,33].…”

Section: Introductionmentioning

confidence: 99%

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On data reduction for dynamic vector bin packing

Bevern¹,

Melnikov²,

Smirnov³

et al. 2022

Preprint

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We study a dynamic vector bin packing (DVBP) problem. We show hardness for shrinking arbitrary DVBP instances to size polynomial in the number of request types or in the maximal number of requests overlapping in time. We also present a simple polynomial-time data reduction algorithm that allows to recover (1 + ε)-approximate solutions for arbitrary ε > 0. It shrinks instances from Microsoft Azure and Huawei Cloud by an order of magnitude for ε = 0.02.

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Section: Problem 35 (2d Vector Bin Packing (2d-vbp))mentioning

confidence: 88%

Section: Problem 35 (2d Vector Bin Packing (2d-vbp))mentioning

confidence: 99%

See 1 more Smart Citation

On data reduction for dynamic vector bin packing

Bevern¹,

Melnikov²,

Smirnov³

et al. 2022

Preprint

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show abstract

Section: Introductionmentioning

confidence: 99%

“…In [19] Ray showed that there is no APTAS for 2VBP, assuming P = NP. 2 In the same work [19], Ray also showed the problem does not admit an asymptotic approximation ratio better than 600 599 , assuming P = NP. The best known asymptotic ratio for 2VBP prior to this work is 1+ln 3 2 +ε ≈ 1.406 for any ε > 0, due to Bansal, Eliáš and Khan [2].…”

Section: Introductionmentioning

confidence: 99%

An Asymptotic $\left(\frac{4}{3}+\varepsilon\right)$-Approximation for the 2-Dimensional Vector Bin Packing Problem

Kulik¹,

Mnich²,

Shachnai³

2022

Preprint

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We study the 2-Dimensional Vector Bin Packing Problem (2VBP), a generalization of classic Bin Packing that is widely applicable in resource allocation and scheduling. In 2VBP we are given a set of items, where each item is associated with a two-dimensional volume vector. The objective is to partition the items into a minimal number of subsets (bins), such that the total volume of items in each subset is at most 1 in each dimension.We give an asymptotic 4 3 + ε -approximation for the problem, thus improving upon the best known asymptotic ratio of 1 + ln 3 2 + ε ≈ 1.406 due to Bansal, Eliáš and Khan [2]. Our algorithm applies a novel Round&Round approach which iteratively solves a configuration LP relaxation for the residual instance and samples a small number of configurations based on the solution for the configuration LP. For the analysis we derive an iteration-dependent upper bound on the solution size for the configuration LP, which holds with high probability. To facilitate the analysis, we introduce key structural properties of 2VBP instances, leveraging the recent fractional grouping technique of Fairstein et al. [12].

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Improved Hardness of Approximation for Geometric Bin Packing

Ray¹,

Sandeep²

2023

Preprint

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The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of d-dimensional rectangles, and the goal is to pack them into unit d-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when d = 2. For general d, the best known approximation algorithm has an approximation guarantee exponential in d, while the best hardness of approximation is still a small constant inapproximability from the case when d = 2. In this paper, we show that the problem cannot be approximated within d 1− factor unless NP=ZPP.Recently, d-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within Ω(log d) when d is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when d is fixed, we prove a couple of key properties of the Geometric Packing Dimension that highlight the difference between Geometric Packing Dimension and Packing Dimension.

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There is no APTAS for 2-dimensional vector bin packing: Revisited

Cited by 3 publications

References 0 publications

On data reduction for dynamic vector bin packing

On data reduction for dynamic vector bin packing

An Asymptotic $\left(\frac{4}{3}+\varepsilon\right)$-Approximation for the 2-Dimensional Vector Bin Packing Problem

Improved Hardness of Approximation for Geometric Bin Packing

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