Approximation and online algorithms for multidimensional bin packing: A survey

Christensen, Henrik I.; Khan, Arindam; Pokutta, Sebastian; Tetali, Prasad

doi:10.1016/j.cosrev.2016.12.001

Cited by 195 publications

(125 citation statements)

References 147 publications

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“…The best known (asymptotic) result for two-dimensional bin packing is due to Bansal and Khan and it is based on this configuration-LP, achieving an approximation ratio of 1.405 [6] which improves a series of previous results [21,5,7,26,10]. See also the recent survey in [9] and [27].…”

Section: Other Related Workmentioning

confidence: 57%

Approximating Geometric Knapsack via L-Packings

Gálvez

Grandoni

Heydrich

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

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We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2 + ε [Jansen and Zhang, SODA 2004]. In this paper we break the 2 approximation barrier, achieving a polynomial-time 17 9 + ε < 1.89 approximation, which improves to 558 325 + ε < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. The items of these two types possibly interact in a complex manner at the corner of the L.The above structural result is not enough however: the best-known approximation ratio for the subproblem in the L-shaped region is 2 + ε (obtained via a trivial reduction to one-dimensional knapsack by considering tall or wide items only). Indeed this is one of the simplest special settings of the problem for which this is the best known approximation factor. As a second major, and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems.We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also in this case the best-known polynomial-time approximation factor (even for the cardinality case) is 2 + ε [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time 3/2 + ε-approximation for 2DKR, which improves to 4/3 + ε in the cardinality case.

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Section: Other Related Workmentioning

confidence: 57%

Approximating Geometric Knapsack via L-Packings

Gálvez

Grandoni

Heydrich

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

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“…The best known offline algorithm for the d-dimensional Hyperrectangle Packing problem has ratio 1.69103 d−1 and is due to Caprara [7] (see also [8]). We can thus use Theorem 7 with γ = 1.69103 d−1 and β = 2 2d −3·2 d +1 2 2d (2 d −1) to conclude the following theorem.…”

Section: D-dimensional Bin Packingmentioning

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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“…We give an overview of related work in offline, online, and sublinear algorithms, and highlight the differences between online and streaming algorithms. Recent surveys of Christensen et al [13] and Coffman et al [14] have a more comprehensive overview.…”

Section: Related Workmentioning

confidence: 99%

“…For general d, a relatively simple algorithm based on an LP relaxation, due to Chekuri and Khanna [11], remains the best known, with an approximation guarantee of 1 + εd + O(log 1 ε ). The problem is APXhard even for d = 2 [40], and cannot be approximated within a factor better than d 1−ε for any fixed ε > 0 [13] if d is arbitrarily large. Hence, our streaming d + ε-approximation for VECTOR BIN PACK-ING asymptotically achieves the offline lower bound.…”

Section: Bin Packingmentioning

confidence: 99%

Streaming Algorithms for Bin Packing and Vector Scheduling

Cormode

Veselý

2020

Lecture Notes in Computer Science

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Problems involving the efficient arrangement of simple objects, as captured by bin packing and makespan scheduling, are fundamental tasks in combinatorial optimization. These are well understood in the traditional online and offline cases, but have been less well-studied when the volume of the input is truly massive, and cannot even be read into memory. This is captured by the streaming model of computation, where the aim is to approximate the cost of the solution in one pass over the data, using small space. As a result, streaming algorithms produce concise input summaries that approximately preserve the optimum value.We design the first efficient streaming algorithms for these fundamental problems in combinatorial optimization. For BIN PACKING, we provide a streaming asymptotic 1 + ε-approximation with O 1 ε memory, where O hides logarithmic factors. Moreover, such a space bound is essentially optimal. Our algorithm implies a streaming d + ε-approximation for VECTOR BIN PACKING in d dimensions, running in space O d ε . For the related VECTOR SCHEDULING problem, we show how to construct an input summary in space O(d 2 · m/ε 2 ) that preserves the optimum value up to a factor of 2 − 1 m + ε, where m is the number of identical machines.

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Approximation and online algorithms for multidimensional bin packing: A survey

Cited by 195 publications

References 147 publications

Approximating Geometric Knapsack via L-Packings

Approximating Geometric Knapsack via L-Packings

Robust Online Algorithms for Certain Dynamic Packing Problems

Streaming Algorithms for Bin Packing and Vector Scheduling

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