A (5/3 + ε)-Approximation for Strip Packing

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

doi:10.1007/978-3-642-22300-6_40

Cited by 16 publications

(9 citation statements)

References 17 publications

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“…This problem is APX-hard 1 (by a reduction from bin packing). Improving on earlier results [17,18], Harren et al [12] recently found a (5/3 + ε)-approximation. Jensen and Solis-Oba [14] devised an AFPTAS 2 which packs the rectangles into a box of height at most (1 + ε)OPT + 1 for every ε > 0.…”

Section: Introductionsupporting

confidence: 62%

Packing anchored rectangles

Dumitrescu

Tóth

2015

Combinatorica

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Let S be a set of n points in the unit square [0, 1] 2 , one of which is the origin. We construct n pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in S, and the rectangles jointly cover at least a positive constant area (about 0.09). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2.

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Section: Introductionsupporting

confidence: 62%

Packing anchored rectangles

Dumitrescu

Tóth

2015

Combinatorica

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“…In terms of approximation schemes, Harren et al (2011) presented a 5/3 + polynomial time approximation scheme. Kenyon and Rémila (2000) proposed an asymptotic fully polynomial time approximation scheme providing a solution of cost not higher than 1 + opt + O 1/ 2 , where opt is the optimal solution value.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Benders' Cuts for the Strip Packing Problem

Côté

Dell’Amico

Iori

2014

Operations Research

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We study the strip packing problem, in which a set of two-dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height, with the aim of minimizing the height used. The problem is important because it models a large number of real-world applications, including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste, scheduling problems in which tasks require a contiguous subset of identical resources, and container loading problems arising in the transportation of items that cannot be stacked one over the other.The strip packing problem has been attacked in the literature with several heuristic and exact algorithms, nevertheless, benchmark instances of small size remain unsolved to proven optimality. In this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort. Our method is based on a Benders' decomposition, in which in the master we cut items into unit-width slices and pack them contiguously in the strip, and in the slave we attempt to reconstruct the rectangular items by fixing the vertical positions of their unit-width slices. If the slave proves that the reconstruction of the items is not possible, then a cut is added to the master, and the algorithm is reiterated.We show that both the master and the slave are strongly -hard problems and solve them with tailored preprocessing, lower and upper bounding techniques, and exact algorithms. We also propose several new techniques to improve the standard Benders' cuts, using the so-called combinatorial Benders' cuts, and an additional lifting procedure. Extensive computational tests show that the proposed algorithm provides a substantial breakthrough with respect to previously published algorithms.

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“…The bottomleft k-local search algorithm starts with an initial bottom-left packing and in each iteration the algorithm tries to permute k rectangles such that the bottom-left algorithm on the new ordering returns a packing with strictly lower height. Firstly, we show a lower bound equal to 2 for the bottom-left k-local search algorithm, implying that this algorithm cannot find an ordering such that the bottom-left algorithm has approximation ratio better than the currently best-known (5/3 + ε)-approximation ratio from [7]. Secondly, we also show that the local search algorithm may need an exponential number of iterations before reaching a local optimum.…”

Section: Introductionmentioning

confidence: 86%

“…Moreover, this reduction establishes that unless P=NP, there cannot exist a (3/2 − ε)-approximation algorithm for Strip Packing. Currently, the bestknown approximation algorithm achieves an approximation ratio of 5/3 + ε [7,5]. However, this algorithm is rather complicated and may not be of practical relevance.…”

Section: Introductionmentioning

confidence: 99%

On the nearest neighbor rule for the metric traveling salesman problem

Hougardy

Mirko

2015

Discrete Applied Mathematics

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We present a very simple family of traveling salesman instances with n cities where the nearest neighbor rule may produce a tour that is Θ(log n) times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time a lower bound in the rectilinear case.keywords: traveling salesman problem; nearest neighbor rule; approximation algorithm

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A (5/3 + ε)-Approximation for Strip Packing

Cited by 16 publications

References 17 publications

Packing anchored rectangles

Packing anchored rectangles

Combinatorial Benders' Cuts for the Strip Packing Problem

On the nearest neighbor rule for the metric traveling salesman problem

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