Bin packing can be solved within 1 + ε in linear time

Véga, W. Fernandez de la; Lueker, George S.

doi:10.1007/bf02579456

Cited by 429 publications

(142 citation statements)

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“…Fernandez de la Vega and Lueker [4] designed an APTAS for standard bin packing. Their work was followed by the work of Karmarkar and Karp [10] who developed an AFPTAS.…”

Section: { A(i) Opt(i) |Opt(i) = N}mentioning

confidence: 99%

Approximation Schemes for Packing Splittable Items with Cardinality Constraints

2010

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We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of items must be packed into as few bins as possible. Items may be split, but each bin may contain at most k (parts of) items, where k is some fixed constant. Complicating the problem further is the fact that items may be larger than 1, which is the size of a bin. We close this problem by providing a polynomial-time approximation scheme for it. We first present a scheme for the case k = 2 and then for the general case of constant k.Additionally, we present dual approximation schemes for k = 2 and constant k. Thus we show that for any ε > 0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1 + ε.

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“…Fernandez de la Vega and Lueker [4] designed an APTAS for standard bin packing. Their work was followed by the work of Karmarkar and Karp [10] who developed an AFPTAS.…”

Section: { A(i) Opt(i) |Opt(i) = N}mentioning

confidence: 99%

Approximation Schemes for Packing Splittable Items with Cardinality Constraints

2010

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“…Two of the most cited works in offline consolidation are: the APTAS algorithm proposed in [13], and the exact algorithm proposed in [14]. Finally, the work presented in [15] proposes a decentralized approach by using ant colony optimization.…”

Section: Offline Consolidationmentioning

confidence: 99%

The positive impact of failures on energy efficient virtual machines consolidation

Gonzalez

Helvik

2014

2014 IEEE International Conference on Communications (ICC)

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Abstract-Failures are undesirable events that should be always prevented. This paper agrees with this concept, but it is also aware that failures are unavoidable events. The main objective of this paper is to show how the failure handling through clever fault management mechanisms produces a direct positive impact on energy efficient cloud computing deployment. We address the virtual machine consolidation problem, and its associated bin packing problem. In spite of being studied for several years, it is, to the authors knowledge, the first time that this is studied under the presence of server failures that force the migration of some virtual machines at unexpected random times. We found that the extra dynamics generated by the compulsory reallocation of virtual machines produces a reduction in the number of servers needed, allowing some servers to be switched off, and hence having a more energy efficient cloud operation. Through some case studies, we show the amount of energy saved by the effect of random servers failures. Finally, we show that under some conditions, the optimal number of active servers may be reached.

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“…2.4, each sub-instance S u can be scheduled separately on a disjoint set of machines by any algorithm for the bin-packing problem. In particular, by Corollary 2.8 one can use the asymptotic PTAS of Fernandez de la Vega and Lueker (1981) that uses at most (1 + ε)OPT(I ) + 1 bins to pack an instance I . This AP-TAS can in fact provide a stronger corollary.…”

Section: An Algorithm That Uses 2(1 + ε)W (I ) + Log W Max Machinesmentioning

confidence: 99%

“…This AP-TAS can in fact provide a stronger corollary. Let SIZE(I ) denote the total size of items to be packed, then the number of bins used in the APTAS in Fernandez de la Vega and Lueker (1981) is at most (1 + ε)SIZE(I ) + 1. In the analysis of our algorithm for arbitrary instances, we are going to use the following stronger corollary of this APTAS for the identical-windows case.…”

Section: An Algorithm That Uses 2(1 + ε)W (I ) + Log W Max Machinesmentioning

confidence: 99%