Probabilistic Analysis of the Dual Next-Fit Algorithm for Bin Covering

Fischer, Carsten; Röglin, Heiko

doi:10.1007/978-3-662-49529-2_35

Cited by 10 publications

(14 citation statements)

References 15 publications

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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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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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show abstract

“…Under random arrival order, Christ et al [6] showed that RR ∞ DNF ≤ 4∕5 . 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: 94%

“…This technique has already been used in [28] to establish the lower bound of 1.08, however, without a formal proof. Apparently, the only published proofs of this connection address bin covering [6,13]. We provide a constructive proof of Lemma 3 in Appendix C for completeness.…”

Section: Asymptotic Random Order Ratiomentioning

confidence: 99%

“…Absolute Random Order RatioTheorem 3 follows from the following lemma. There exists a list I such that E[BF(I )] =13 10 OPT(I).Proof Let > 0 be sufficiently small and let I ∶= (a 1 , a 2 , b 1 , b 2 , c) whereAn optimal packing of I has two bins {a 1 , a 2 , c} and {b 1 , b 2 } , thus OPT(I) = 2 . Subsequently, we argue that Best Fit needs two or three bins depending on the order of arrival.…”

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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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“…Random-order analysis has also been applied to other problems, e.g., knapsack [Babaioff et al 2007], bipartite matching [Goel and Mehta 2008;Devanur and Hayes 2009], scheduling [Osborn and Torng 2008;Göbel et al 2015], bin covering [Christ et al 2014;Fischer and Röglin 2016], and facility location [Meyerson 2001]. However, the analysis is often rather challenging, and in [Coffman Jr. et al 2008], a simplified version of the random-order ratio is used for bin packing.…”

Section: Introductionmentioning

confidence: 99%

Relative Worst-order Analysis

2021

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The standard measure for the quality of online algorithms is the competitive ratio. This measure is generally applicable, and for some problems it works well, but for others it fails to distinguish between algorithms that have very different performance. Thus, ever since its introduction, researchers have worked on improving the measure, defining variants, or defining measures based on other concepts to improve on the situation. Relative worst-order analysis (RWOA) is one of the most thoroughly tested such proposals. With RWOA, many separations of algorithms not obtainable with competitive analysis have been found. In RWOA, two algorithms are compared directly, rather than indirectly as is done in competitive analysis, where both algorithms are compared separately to an optimal offline algorithm. If, up to permutations of the request sequences, one algorithm is always at least as good and sometimes better than another, then the first algorithm is deemed the better algorithm by RWOA. We survey the most important results obtained with this technique and compare it with other quality measures. The survey includes a quite complete set of references.

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Probabilistic Analysis of the Dual Next-Fit Algorithm for Bin Covering

Cited by 10 publications

References 15 publications

Online Bin Covering with Limited Migration

Online Bin Covering with Limited Migration

Best Fit Bin Packing with Random Order Revisited

Relative Worst-order Analysis

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