A quasi-PTAS for the Two-Dimensional Geometric Knapsack Problem

Adamaszek, Anna; Wiese, Andreas

doi:10.1137/1.9781611973730.98

Cited by 29 publications

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“…The maximum area independent set (MAIS) problem for rectangles (squares, or disks, etc. ), is that of selecting a maximum area packing from a given set [3]; see classic papers such as [6,32,33,34,35] and also more recent ones [10,11,21] for quantitative bounds and constant approximations. These optimization problems are NP-hard, and there is a rich literature on approximation algorithms.…”

Section: Introductionmentioning

confidence: 99%

Anchored rectangle and square packings

Balas

Dumitrescu

Tóth

2017

Discrete Optimization

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For points p 1 , . . . , p n in the unit square [0,1] 2 , an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r 1 , . . . , r n ⊆ [0, 1] 2 such that point p i is a corner of the rectangle r i (that is, r i is anchored at p i ) for i = 1, . . . , n. We show that for every set of n points in [0,1] 2 , there is an anchored rectangle packing of area at least 7/12 − O(1/n), and for every n ∈ N, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 − ε for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n O(1/ε) and exp(poly(log(n/ε))) time, respectively.

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Section: Introductionmentioning

confidence: 99%

Anchored rectangle and square packings

Balas

Dumitrescu

Tóth

2017

Discrete Optimization

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“…The PTAS in [13] for geometric knapsack for squares (and many other algorithms for geometric problems, e.g., [2,16,3]) applies the mentioned shifting technique to distinguish squares into large and small squares. Eventually, it is shown that there is a (1 + )-approximative packing which is structured into O (1) large squares and O (1) cells.…”

Section: Our Techniquesmentioning

confidence: 99%

“…The running time of all these (1 + )-approximation algorithms is Ω(n 1/ 1/ ). Moreover, for quasi-polynomially bounded input there is even a (1 + )-approximation in quasi-polynomial time [2].…”

Section: Other Related Workmentioning

confidence: 99%

“…We treat the rectangles in M separately using the following lemma and pack them into the additional space gained by increasing the size of the knapsack (a similar argumentation was used in [2]). The intuition is that increasing the size of the knapsack by a factor 1 + O( ) gains free space that is by a constant factor larger than the total space needed for the medium rectangles.…”

Section: Rectangle Classificationmentioning

confidence: 99%

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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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“…There, given a collection of rectangular items I, where each item i ∈ I is defined by its width w i ∈ N, height h i ∈ N, and weight ω i ∈ R + , and a rectangular box of size W × H, the goal is to pack a subcollection of items of maximum total weight into the given box so that no two items overlap. For this problem there is a (2 + )-approximation algorithm by Jansen and Zhang [13], and a QPTAS by Adamaszek and Wiese [2].…”

Section: Introductionmentioning

confidence: 99%

Hardness of Approximation for Strip Packing

Adamaszek

Kociumaka

Pilipczuk

et al. 2017

ACM Trans. Comput. Theory

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Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, e.g. in scheduling and stock-cutting, and has been studied extensively.When the dimensions of objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than 3/2, unless P = NP. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a (1.4 + ) approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values in the input data are allowed. Their result has subsequently been improved to a (4/3 + ) approximation by two independent research groups [FSTTCS 2016, arXiv:1610]. This raises a question whether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related two-dimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack.In this paper we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than 12/11, even when admitting only polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP ⊆ DTIME(2 polylog(n) ).

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A quasi-PTAS for the Two-Dimensional Geometric Knapsack Problem

Cited by 29 publications

References 0 publications

Anchored rectangle and square packings

Anchored rectangle and square packings

Faster approximation schemes for the two-dimensional knapsack problem

Hardness of Approximation for Strip Packing

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