A Tight (3/2+ε) Approximation for Skewed Strip Packing

Gálvez, Waldo; Grandoni, Fabrizio; Ameli, Afrouz Jabal; Jansen, Klaus; Khan, Arindam; Rau, Malin

doi:10.4230/lipics.approx/random.2020.44

Cited by 3 publications

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“…Geometric Bin Packing with Skewed Items (for a class of algorithms called rounding based algorithms) are skewed instances, where one dimension is 1 − ε and the other dimension is ε. Galvez el al. [18] recently studied strip packing when all items are skewed. For skewed instances, they showed (3/2 − ε) hardness of approximation and a matching (3/2 + ε)-approximation algorithm.…”

Section: :4mentioning

confidence: 99%

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

Khan

Sharma

2023

Algorithmica

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In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let λ be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by λ opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ≤ λ ≤ 1.692. Bansal and Khan [SODA'14] conjectured that λ = 4/3. The conjecture, if true, will imply a (4/3 + ε)-approximation algorithm for 2BP. According to convention, for a given constant δ > 0, a rectangle is large if both its height and width are at least δ, and otherwise it is called skewed. We make progress towards the conjecture by showing λ = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on λ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS.

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Section: :4mentioning

confidence: 99%

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

Khan

Sharma

2023

Algorithmica

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“…For VP, hard instances are again skewed, e.g., Bansal, Eliáš and Khan [6] showed that hard instances for 2-D VP (for a class of algorithms called rounding based algorithms) are skewed instances, where one dimension is 1 − ε and the other dimension is ε. Galvez el al. [17] recently studied strip packing when all items are skewed. For skewed instances, they showed (3/2 − ε) hardness of approximation and a matching (3/2 + ε)-approximation algorithm.…”

Section: Related Workmentioning

confidence: 99%

“…Skewed instances are important in geometric packing (see Section 1.1). This special case is practically relevant [17]: e.g., in scheduling, it captures scenarios where no job can consume a significant amount of a shared resource (energy, memory space, etc.) for a significant amount of time.…”

Section: Introductionmentioning

confidence: 99%

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

Khan¹,

Sharma²

2021

Preprint

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In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let λ be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by λ opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ≤ λ ≤ 1.692. Bansal and Khan [SODA'14] conjectured that λ = 4/3. The conjecture, if true, will imply a (4/3+ε)-approximation algorithm for 2BP. According to convention, for a given constant δ > 0, a rectangle is large if both its height and width are at least δ, and otherwise it is called skewed. We make progress towards the conjecture by showing λ = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on λ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS.

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“…It is not even known whether 2-D GKS is APX-hard or not. There are many other related important geometric packing problems, such as strip packing [21,19] and maximum independent set of rectangles [31]. For surveys on multidimensional packing, see [12,30].…”

Section: Related Workmentioning

confidence: 99%

Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

Khan,

Sharma,

Sreenivas

2021

Preprint

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We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i th item has width w(i), height h(i), and d nonnegative weights v 1 (i), v 2 (i), . . . , v d (i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j th weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well studied in practice, surprisingly, approximation algorithms for this problem have rarely been explored.We first obtain two simple algorithms for GVBP having asymptotic approximation ratios 6(d + 1) and 3(1 + ln(d + 1) + ε). We then extend the Round-and-Approx (R&A) framework [3,6] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of 2(1 + ln((d + 4)/2)) + ε for GVBP, which improves to 2.919 + ε for the special case of d = 1. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids.

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A Tight (3/2+ε) Approximation for Skewed Strip Packing

Cited by 3 publications

References 0 publications

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items

Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

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