Approximation Algorithms for 3D Orthogonal Knapsack

Diedrich, Florian; Harren, Rolf; Jansen, Klaus; Thöle, Ralf; Thomas, Henning

doi:10.1007/s11390-008-9170-7

Cited by 17 publications

(12 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The biggest tasks are then packed first and the smallest ones last, into the holes left by the bigger ones. This is similar to the classical knapsack problem [49], and it can be expected that optimal packing is a decisive ingredient also in global cluster structure optimisation, quite independent of the inter-particle potential(s). Therefore, we make use of the same idea and order the building blocks of the cluster by their size and pack them as described above.…”

Section: Program Designmentioning

confidence: 74%

OGOLEM: Global cluster structure optimisation for arbitrary mixtures of flexible molecules. A multiscaling, object-oriented approach

Dieterich¹,

Hartke²

2010

Molecular Physics

117

View full text Add to dashboard Cite

In practical applications, global cluster structure optimisation has so far been limited largely to homogeneous clusters of atoms or small molecules, with little or no choice in the calculation of inter-particle forces. We eliminate these limitations by presenting a new program suite OGOLEM that is universal by design, both in cluster composition (including arbitrarily heterogeneous clusters of complicated molecules) and in its interfaces to force calculation backends. This is demonstrated by exemplary applications in two novel fields: strongly heterogeneous Lennard-Jones clusters (ternary, quaternary, quinary) and mixed clusters of the aminoglycoside Kanamycin A with sodium cations.

show abstract

Section: Program Designmentioning

confidence: 74%

OGOLEM: Global cluster structure optimisation for arbitrary mixtures of flexible molecules. A multiscaling, object-oriented approach

Dieterich¹,

Hartke²

2010

Molecular Physics

117

View full text Add to dashboard Cite

show abstract

“…We used Steinberg's algorithm when d g = 2. When d g = 3, we can use the algorithm of [15,Section 2] to pack J into at most 9 bins. For d g > 3, we can use a variant of the HDH 4 algorithm [10] to pack J into at most 4 dg + 2 dg bins (details in Appendix C).…”

Section: Simple Algorithmsmentioning

confidence: 99%

Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

Khan,

Sharma,

Sreenivas

2021

Preprint

View full text Add to dashboard Cite

We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i th item has width w(i), height h(i), and d nonnegative weights v 1 (i), v 2 (i), . . . , v d (i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j th weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well studied in practice, surprisingly, approximation algorithms for this problem have rarely been explored.We first obtain two simple algorithms for GVBP having asymptotic approximation ratios 6(d + 1) and 3(1 + ln(d + 1) + ε). We then extend the Round-and-Approx (R&A) framework [3,6] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of 2(1 + ln((d + 4)/2)) + ε for GVBP, which improves to 2.919 + ε for the special case of d = 1. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids.

show abstract

“…They were able to solve in a small computational time instances of the OR-Library for which no optimal solution was known. Diedrich et al (2008) proposed approximation algorithms for the 3DK problem with approximation ratios (9 + ), (8 + ) and (7 + ); and for the 3DK r problem they designed an approximation algorithm with ratio (5 + ).…”

Section: Three-dimensional Strip Packing Problem (3sp)mentioning

confidence: 99%

Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing

Queiroz

Miyazawa

Wakabayashi

et al. 2012

Computers & Operations Research

View full text Add to dashboard Cite

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: Unbounded Knapsack, Cutting Stock and Strip Packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the Unbounded 3D Knapsack problem, we extend the recurrence formula proposed by Beasley for the Rectangular Knapsack Problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the Unbounded Knapsack problem in which the cuts must be staged. For the 3D Cutting Stock problem and its variants in which the bins have different sizes (and the cuts must be staged), we present column generation based algorithms. Modified versions of the algorithms for the 3D Cutting Stock problems with stages are then used to build algorithms for the 3D Strip Packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of moderate size.

show abstract

Approximation Algorithms for 3D Orthogonal Knapsack

Cited by 17 publications

References 28 publications

OGOLEM: Global cluster structure optimisation for arbitrary mixtures of flexible molecules. A multiscaling, object-oriented approach

OGOLEM: Global cluster structure optimisation for arbitrary mixtures of flexible molecules. A multiscaling, object-oriented approach

Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

Algorithms for 3D guillotine cutting problems: Unbounded knapsack, cutting stock and strip packing

Contact Info

Product

Resources

About