Probabilistic Analysis of Online (Class-Constrained) Bin Packing and Bin Covering

Fischer, Carsten; Röglin, Heiko

doi:10.1007/978-3-319-77404-6_34

Cited by 7 publications

(12 citation statements)

References 13 publications

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“…Coffman et al [9] analyzed next-fit in the random order model and showed that RR ∞ NF = 2 , matching the asymptotic approximation ratio RR ∞ NF = 2 (see Table 1). Fischer and Röglin [14] obtained analogous results for Worst Fit [24] and Smart Next Fit [34]. Therefore, all three algorithms fail to perform better in the random order model than in the adversarial model.…”

Section: Bin Packing Kenyon Introduced the Notion Of Asymptotic Random Order Ratio Rr ∞mentioning

confidence: 75%

“…This upper bound was improved later by Fischer and Röglin [13] to RR ∞ DNF ≤ 2∕3 . The same group of authors further showed that RR ∞ DNF ≥ 0.501 , i.e., DNF performs strictly better under random order than in the adversarial setting [14].…”

Section: Bin Packing Kenyon Introduced the Notion Of Asymptotic Random Order Ratio Rr ∞mentioning

confidence: 91%

“…Murgolo [33] showed that next-fit is monotone, while Best Fit and First Fit are not monotone in general. The concept of monotonicity also arises in related optimization problems, such as scheduling [18] and bin covering [14].…”

Section: Bin Packing Kenyon Introduced the Notion Of Asymptotic Random Order Ratio Rr ∞mentioning

confidence: 99%

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Best Fit Bin Packing with Random Order Revisited

2021

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Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

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Section: Bin Packing Kenyon Introduced the Notion Of Asymptotic Random Order Ratio Rr ∞mentioning

confidence: 75%

Section: Bin Packing Kenyon Introduced the Notion Of Asymptotic Random Order Ratio Rr ∞mentioning

confidence: 91%

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Best Fit Bin Packing with Random Order Revisited

2021

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“…This was improved to an asymptotic fully polynomial time approximation scheme (AFPTAS) by Jansen and Solis-Oba in [21]. Many different variants of this problem have also been investigated: If a certain number of classes needs to be part of each bin [10,16]; if items are drawn probabilistically [15,16]; if bins have different sizes [8,24]; if the competitiveness is not measured with regard to an optimal offline algorithm [5,9]. More variants are discussed for example in [18] and lower bounds for several variants are studied in [2].…”

Section: Known Results For Bin Coveringmentioning

confidence: 99%

Online Bin Covering with Limited Migration

Berndt¹,

Epstein²,

Jansen³

et al. 2019

Preprint

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Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor β: When an object o of size s(o) arrives, the decisions for objects of total size at most β • s(o) may be revoked. Usually β should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classic problems such as bin packing or makespan minimization. The dual of makespan minimization -the Santa Claus or machine covering problem -has also been studied, whereas the dual of bin packing -the bin covering problem -has not been looked at from such a perspective.In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case -where only insertions are allowed -and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small ε). We therefore resolve the competitiveness of the bin covering problem with migration.

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“…Additional algorithmic results established for the bin covering problem and variants of it appear in e.g. [7,8,25,12,13,19,6,21,16,17,3,5].…”

Section: Opt(i) Alg(i)mentioning

confidence: 99%

The near exact bin covering problem

Levin¹

2022

Preprint

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We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant ∆, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and 1 + ∆. Our goal is to maximize the number of near exact covered bins. If ∆ = 0 or ∆ > 0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that P = N P ). However, for the case where ∆ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

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Probabilistic Analysis of Online (Class-Constrained) Bin Packing and Bin Covering

Cited by 7 publications

References 13 publications

Best Fit Bin Packing with Random Order Revisited

Best Fit Bin Packing with Random Order Revisited

Online Bin Covering with Limited Migration

The near exact bin covering problem

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