New Approximability Results for 2-Dimensional Packing Problems

Jansen, Klaus; Solis-Oba, Roberto

doi:10.1007/978-3-540-74456-6_11

Cited by 29 publications

(37 citation statements)

References 14 publications

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“…However, there are PTASs for the resource augmentation setting in both [9] or in only one [12] dimension, and for the case that the profit of each item equals its area [3]. The running time of all these (1 + )-approximation algorithms is Ω(n 1/ 1/ ).…”

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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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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“…This is an absolute performance bound, i.e., the height achieved is at most 5/3 + ε times the optimal height. Many other authors have proposed algorithms with asymptotic performance guarantees [4,13,11] where an additive term is allowed.…”

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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“…Kenyon & Rémila [9] and Jansen & van Stee [7] gave asymptotic FPTAS's for the problem without rotations and with rotations, respectively. The additive constant of these algorithms was recently improved from O(1/ε 2 ) to 1 by Jansen & Solis-Oba [6]. Thus, most versions of the strip packing problem are now closed.…”

Section: Introductionmentioning

confidence: 99%

“…Our main lemma on the packability of certain sets of small items is of independent interest. We started our investigation on the problem with an algorithm of Jansen & Solis-Oba [6] that finds a packing of profit (1−δ)OPT into a bin of size (1, 1+δ), where OPT denotes the optimum for packing into a unit bin. Using the area of the items as their profit gives an algorithm that packs almost everything into an δ-augmented bin.…”

Section: Introductionmentioning

confidence: 99%