A<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-approximation for strip packing

Harren, Rolf; Jansen, Klaus; Prädel, Lars; Stee, Rob van

doi:10.1016/j.comgeo.2013.08.008

Cited by 52 publications

(36 citation statements)

References 12 publications

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“…Independently, Schiermeyer [21] and Steinberg [24] presented a 2-approximation algorithm. The current championship is due to Harren [13] with an approximation algorithm of 5/3 + ε, where ε is any given small positive number. Another important result is an asymptotic PTAS which was given by Kenyon and Remila [18], and the additional constant was improved by Jansen and Solis-Oba [16].…”

Section: Related Workmentioning

confidence: 99%

Coordination Mechanisms for Selfish Parallel Jobs Scheduling

Zhang

2012

Lecture Notes in Computer Science

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Abstract.A set of parallel jobs must be scheduled in a grid, which has multi clusters that consist of many identical processors, to minimize the global objective function, the makespan. A parallel job requires several processors for simultaneously executing and it needs some unit times to finish its execution. In practice, each parallel job is owned by an independent agent, which is selfish and select a cluster to minimize its own completion time. This scenario can be represented as a coordination mechanism game, in which the players are job owners, and the strategies are the clusters, and the player's disutility is the completion time of its job in the corresponding schedule.In this work, we design and analyze coordination mechanisms for machines, which aim to minimize the price of anarchy. We study two classes of scheduling policies, the Bottom-Left based algorithms and the ShelfPacking based algorithms, both in a homogeneous grid and in a heterogeneous grid. We derive upper and lower bounds on the price of anarchy of these coordination mechanisms. We also show that such games are potential games that converge in a linear number of rounds.

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Section: Related Workmentioning

confidence: 99%

Coordination Mechanisms for Selfish Parallel Jobs Scheduling

Zhang

2012

Lecture Notes in Computer Science

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“…The 2SP problem is very well-studied [14], and generalizes the bin packing problem, which is equivalent to the case in which all rectangles have unit height. The current best approximation algorithm for 2SP has an approximation factor of 5/3 + ε for any ε > 0 [9], and was achieved after a long sequence of successive improvements [1,16,17,19]. This is an absolute performance bound, i.e., the height achieved is at most 5/3 + ε times the optimal height.…”

Section: Introductionmentioning

confidence: 99%

Smart-Grid Electricity Allocation via Strip Packing with Slicing

Alamdari

Biedl

Chan

et al. 2013

Lecture Notes in Computer Science

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Abstract. One advantage of smart grids is that they can reduce the peak load by distributing electricity-demands over multiple short intervals. Finding a schedule that minimizes the peak load corresponds to a variant of a strip packing problem. Normally, for strip packing problems, a given set of axis-aligned rectangles must be packed into a fixed-width strip, and the goal is to minimize the height of the strip. The electricityallocation application can be modelled as strip packing with slicing: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions in which a vertical line intersects two slices of the same rectangle. We give a fully polynomial time approximation scheme for this problem, as well as a practical polynomial time algorithm that slices each rectangle at most once and yields a solution of height at most 5/3 times the optimal height.

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“…For this problem, an absolute approximation guarantee of 5/3 + can be achieved, while a lower bound of 3/2 is known. [10]. Both of these results use the doublyexponential algorithm in [3] as a subroutine.…”

Section: Other Related Workmentioning

confidence: 99%

Faster approximation schemes for the two-dimensional knapsack problem

Heydrich¹,

Wiese²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + )-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1/ , i.e., Ω(n 2 2 1/ ). A double or triple exponential dependence on 1/ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ )-approximation algorithm with a running time of O (1) · n O(1) . In particular, the exponent of n in the running time does not depend on at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(2 2 1/ ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n 2 2 1/ ) and we improve both to O (1) · n O(1) . We complete our result by giving an algorithm for twodimensional knapsack for rectangles under (1 + )-resource augmentation. In this setting, we also improve the best known running time of Ω(n 1/ 1/ ) to O (1) · n O(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well.

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A(5/3+ε)-approximation for strip packing

Cited by 52 publications

References 12 publications

Coordination Mechanisms for Selfish Parallel Jobs Scheduling

Coordination Mechanisms for Selfish Parallel Jobs Scheduling

Smart-Grid Electricity Allocation via Strip Packing with Slicing

Faster approximation schemes for the two-dimensional knapsack problem

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