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### On Weighted Rectangle Packing with Large Resources [chapter]

Aleksei V. Fishkin, Olga Gerber, Klaus Jansen
IFIP International Federation for Information Processing
We study the problem of packing a set of Ò rectangles with weights into a dedicated rectangle so that the weight of the packed rectangles is maximized.  ...  We consider the case of large resources, that is, the side length of all rectangles is at most ½ and the side lengths of the dedicated rectangle differ by a factor of at least ½ , for a fixed positive  ...  We can conclude with the following result. Ì ÓÖ Ñ ½½ The algorithm -Bins is a´¾· µ-approximation algorithm. Its running time is polynomial in the number of rectangles Ò for any fixed ¼.  ...

### Approximation Algorithms for Rectangle Packing Problems (PhD Thesis) [article]

Salvatore Ingala
2017 arXiv   pre-print
We revisit some classical results on SP and 2DGK, by proposing a framework based on smaller containers that are packed with simpler rules; while variations of this scheme are indeed a standard technique  ...  In rectangle packing problems we are given the task of placing axis-aligned rectangles in a given plane region, so that they do not overlap with each other.  ...  One key reason is that in the weighted case we cannot discard large items since even one such item might contribute a large fraction to the optimal profit.  ...

### Page 819 of The Journal of the Operational Research Society Vol. 49, Issue 8 [page]

1998 The Journal of the Operational Research Society
We assume that the boxes are available in large quantities and are orthogonally loaded on each pallet (that is, with their sides parallel to the pallet sides).  ...  (faces) into the large rectangle (L, W) (pallet surface) without overlapping, and (ii) the one-dimen- sional problem of stacking the most valuable layers along the width H of the pallet, where the value  ...

### Approximating Geometric Knapsack via L-packings [article]

Waldo Gálvez and Fabrizio Grandoni and Sandy Heydrich and Salvatore Ingala and Arindam Khan and Andreas Wiese
2017 arXiv   pre-print
We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary  ...  Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy.  ...  Is there a PTAS for L-packings with rotations? Our improved approximation algorithms for 2DKR are indeed based on a different approach. Is there a PTAS for O(1) instances of L-packing?  ...

### On the Two-Dimensional Knapsack Problem for Convex Polygons

Arturo Merino, Andreas Wiese, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming
Also, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation, i.e., we allow to increase the size of the knapsack by a factor of 1+δ for some δ  ...  > 0 but compare ourselves with the optimal solution for the original knapsack.  ...  For the same setting, a PTAS is known under resource augmentation in one dimension  and a polynomial time algorithm computing a solution with optimum weight under resource augmentation in both dimensions  ...

### Applying Adaptive Algorithms to Epistatic Domains

Lawrence Davis
1985 International Joint Conference on Artificial Intelligence
John Holland has shown that when adaptive algorithms are used to search certain kinds of extremely large problem spaces, they will converge on a "good" solution fairly quickly.  ...  These techniques are described for two-dimensional bin-packing problems, and summarized for graph coloring problems.  ...  It seems that attempts to combine part of one such solution with parts of another will nearly always result in a worse solution, because the result will be a packing that contains overlaps or large gaps  ...

### Resource Request Mapping Techniques for OFDMA Networks [chapter]

2014 Resource Management in Mobile Computing Environments
OFDMA defines rectangular resource allocation of time slots and frequency carriers, separating in this way the channel into multiple subcarriers.  ...  Subsequently in the following sections are presented analysis and design of various Bin packing algorithms developed in our simulator.  ...  In these cases, a resource (one or more dimensional) is given and an amount of items which need to be packed there is defined.  ...

### On the complexity of sequential rectangle placement in IEEE 802.16/WiMAX systems

Amos Israeli, Dror Rawitz, Oran Sharon
2008 Information and Computation
As far as we know this is the first paper that considers this sequential rectangle placement problem.  ...  We study the problem of scheduling transmissions on the downlink of IEEE 802.16/WiMAX systems that use the OFDMA technology.  ...  The goal is to pack a subset of the rectangles into R such that the total profit of packed rectangles is maximized. The packed rectangles may not overlap.  ...

### Page 3785 of Mathematical Reviews Vol. , Issue 84i [page]

1984 Mathematical Reviews
One can pack squares into a ‘arge rectangle in many ways.  ...  Authors’ summary: “Many problems, such as cutting stock problems and the scheduling of tasks with a shared resource, can be viewed as two-dimensional bin packing problems.  ...

### Rectangle packing with one-dimensional resource augmentation

Klaus Jansen, Roberto Solis-Oba
2009 Discrete Optimization
Recently, Fishkin et al.  proposed an algorithm for the so-called rectangle packing problem with resource augmentation, that packs a subset of rectangles with profit at least (1 − )OPT into an augmented  ...  For our rectangle packing problem we cannot round the width and length of the large rectangles as this would require us to increase both dimensions of the bin.  ...  As for the short rectangles, we use an algorithm for packing rectangles with large resources described in  to select (and actually pack) the short rectangles that will be placed in the containers  ...

### On Guillotine Separable Packings for the Two-dimensional Geometric Knapsack Problem [article]

Arindam Khan, Arnab Maiti, Amatya Sharma, Andreas Wiese
2021 arXiv   pre-print
The goal is to find a (non-overlapping axis-aligned) packing of a maximum profit subset of rectangles into the knapsack.  ...  Our main technical contribution is a structural lemma which shows that any guillotine packing can be converted into another structured guillotine packing with almost the same profit.  ...  E One Sided Resource Augmentation In this section we show that the proof techniques used in  for packing rectangles with resource augmentation maintain the guillotine separability of the rectangles  ...

### Approximation Algorithms for ROUND-UFP and ROUND-SAP [article]

Debajyoti Kar, Arindam Khan, Andreas Wiese
2022 arXiv   pre-print
We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath.  ...  In ROUND-SAP, the tasks are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below  ...  In particular, a feasible solution to the configuration LP for the rectangles in R large is to select one more round for each configuration C with x * C > 0.  ...

### Ant Colony Algorithm for the Weighted Item Layout Optimization Problem [article]

Yi-Chun Xu, Fang-Min Dong, Yong Liu, Ren-Bin Xiao, Martyn Amos
2010 arXiv   pre-print
Two constructive heuristics are proposed, one for packing circular items and the other for packing rectangular items.  ...  This paper discusses the problem of placing weighted items in a circular container in two-dimensional space.  ...  Tests on Weighted Rectangle Layout We use the 4 instances in (Xu et al., 2007a) to test the ACO for WRL, with 5, 6, 9, 20 rectangles respectively.  ...

### Approximation Algorithms for Generalized Multidimensional Knapsack [article]

Arindam Khan and Eklavya Sharma and K. V. N. Sreenivas
2021 arXiv   pre-print
The goal is to find a non-overlapping axis-parallel packing of a subset of items into the given knapsack such that the vector constraints are not violated, i.e., the sum of weights of all the packed items  ...  The input is a set of rectangular items, each with an associated profit and d nonnegative weights (d-dimensional vector), and a square knapsack.  ...  One might also be interested in studying a further generalization of the problem, the (d G , d V ) Knapsack where items are d G dimensional hyper-rectangles with weights in d V dimensions.  ...

### Packing resizable items with application to video delivery over wireless networks

Sivan Albagli-Kim, Leah Epstein, Hadas Shachnai, Tami Tamir
2014 Theoretical Computer Science
Our results are derived through non-standard use of the harmonic technique, as well as by using a transformation of PRI to the problem of covering a region by sliceable rectangles, which finds applications  ...  For general instances we give an algorithm that packs 2 3 OP T (I) − 3 items, where OP T (I) is the number of items packed in an optimal solution for an instance I.  ...  or that the set of all items of X together with ⌊B − w(X)⌋ additional items (that can always be packed), has a sufficiently large number of items.  ...
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