Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/a-tight-3-2-documentclass-12pt-minimal-usepackage-amsmath-usepackage-oB6hU2CEgf
References (53)
-
W Gálvez, F Grandoni, S Ingala, S Heydrich, A Khan, A Wiese (2021)
Approximating geometric knapsack via L-packingsACM Trans. Algorithms, 17
-
disc canbepackedintoonehorizontalcontainerofsize max R ∈ H { w( R ) }× (ε (cid:3) ) 2 H and one ε (cid:3) -area container of size ε (cid:3) W × ε (cid:3) H -
Anna Adamaszek, Andreas Wiese (2013)
Approximation Schemes for Maximum Weight Independent Set of Rectangles2013 IEEE 54th Annual Symposium on Foundations of Computer Science
-
F. Miyazawa, Yoshiko Wakabayashi (2004)
Packing Problems with Orthogonal Rotations -
K. Jansen, Guochuan Zhang (2007)
Maximizing the Total Profit of Rectangles Packed into a RectangleAlgorithmica, 47
-
Waldo G'alvez, F. Grandoni, Arindam Khan, Diego Ram'irez-Romero, Andreas Wiese (2021)
Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and MoreArXiv, abs/2103.10406
-
We describe a packing of T ∪ S hslice with height at most ( 3 / 2 + O (ε)) O PT (see Sect.3.1) into O ε ( 1 ) boxes. This packing leaves a free space of at least ( 1 / 2 + (cid:8)(ε)) O PT + a ( V ) -
H cont can be packed into a set of at most K (cid:3) = O ε (cid:3) ( 1 ) horizontal and ε (cid:3) -area containers, where each container is fully contained in some box from B -
Thesizesoftheabovecontainersbelongtoasetthatcanbecomputedinpolynomialtime -
D. Sleator (1980)
A 2.5 Times Optimal Algorithm for Packing in Two DimensionsInf. Process. Lett., 10
-
Giorgi Nadiradze, Andreas Wiese (2016)
On approximating strip packing with a better ratio than 3/2 -
K. Jansen, M. Rau (2016)
Improved Approximation for Two Dimensional Strip Packing with Polynomial Bounded WidthArXiv, abs/1610.04430
-
Anna Adamaszek, Andreas Wiese (2015)
A quasi-PTAS for the Two-Dimensional Geometric Knapsack Problem -
E. Coffman, John Bruno (1976)
Computer and job-shop scheduling theory -
Waldo Gálvez, F. Grandoni, Salvatore Ingala, A. Khan (2018)
Improved Pseudo-Polynomial-Time Approximation for Strip Packing -
It is possible to pack an extra rectangle ( free box ) of size ε 2 W × 12 O PT into the strip without increasing its final height -
leading to at most (K/ε ) (K/ε ) O(1) containers. By the above construction, the sizes of the containers belong to a set that can be -
E. Coffman, J. Csirik, G. Galambos, S. Martello, D. Vigo (2013)
Bin packing approximation algorithms: Survey and classification -
H. Christensen, A. Khan, S. Pokutta, P. Tetali (2017)
Approximation and online algorithms for multidimensional bin packing: A surveyComput. Sci. Rev., 24
-
A. Khan, M. Pittu (2020)
On Guillotine Separability of Squares and Rectangles -
B. Baker, E. Coffman, R. Rivest (1980)
Orthogonal Packings in Two DimensionsSIAM J. Comput., 9
-
K. Jansen, R. Stee (2005)
On strip packing With rotations -
Parinya Chalermsook, Julia Chuzhoy (2009)
Maximum independent set of rectangles -
K Jansen, M Rau (2019)
Improved approximation for two dimensional strip packing with polynomial bounded widthTheor. Comput. Sci., 789
-
To achieve the above result -
Rolf Harren, K. Jansen, Lars Prädel, R. Stee (2011)
A (5/3 + ε)-approximation for strip packing -
Timothy Chan, Sariel Har-Peled (2009)
Approximation Algorithms for Maximum Independent Set of Pseudo-DisksDiscrete & Computational Geometry, 48
-
TM Chan, S Har-Peled (2012)
Approximation algorithms for maximum independent set of pseudo-disksDiscrete Comput. Geom., 48
-
M. Garey, David Johnson (1978)
`` Strong '' NP-Completeness Results: Motivation, Examples, and ImplicationsJ. ACM, 25
-
M. Karbasioun, G. Shaikhet, E. Kranakis, I. Lambadaris (2013)
Power strip packing of malleable demands in smart grid2013 IEEE International Conference on Communications (ICC)
-
S. Henning, K. Jansen, M. Rau, Lars Schmarje (2017)
Complexity and Inapproximability Results for Parallel Task Scheduling and Strip PackingTheory of Computing Systems, 64
-
Shaojie Tang, Qiuyuan Huang, Xiangyang Li, D. Wu (2013)
Smoothing the energy consumption: Peak demand reduction in smart grid2013 Proceedings IEEE INFOCOM
-
L. Epstein, R. Stee (2004)
This side up!ACM Trans. Algorithms, 2
-
N. Bansal, A. Khan (2014)
Improved Approximation Algorithm for Two-Dimensional Bin Packing -
Given any fixed ordering of T in non-increasing order of height, T can be partitioned into subsequences each one fitting in precisely one vertical container -
David Johnson (2008)
Bin Packing -
Rolf Harren, R. Stee (2009)
Improved Absolute Approximation Ratios for Two-Dimensional Packing Problems -
E. Coffman, M. Garey, David Johnson, R. Tarjan (1980)
Performance Bounds for Level-Oriented Two-Dimensional Packing AlgorithmsSIAM J. Comput., 9
-
K. Jansen, Roberto Solis-Oba (2007)
New Approximability Results for 2-Dimensional Packing Problems -
Arindam Khan, Arnab Maiti, Amatya Sharma, Andreas Wiese (2021)
On Guillotine Separable Packings for the Two-dimensional Geometric Knapsack ProblemArXiv, abs/2103.09735
-
I. Schiermeyer (1994)
Reverse-Fit: A 2-Optimal Algorithm for Packing Rectangles -
Claire Mathieu, É. Rémila (2000)
A Near-Optimal Solution to a Two-Dimensional Cutting Stock ProblemMath. Oper. Res., 25
-
A. Khan (2015)
Approximation algorithms for multidimensional bin packing -
Waldo Gálvez, F. Grandoni, Sandy Heydrich, Salvatore Ingala, A. Khan, Andreas Wiese (2017)
Approximating Geometric Knapsack via L-Packings2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
-
N. Bansal, A. Caprara, K. Jansen, Lars Prädel, M. Sviridenko (2009)
A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability -
A. Steinberg (1997)
A Strip-Packing Algorithm with Absolute Performance Bound 2SIAM J. Comput., 26
-
N. Bansal, J. Correa, Claire Mathieu, M. Sviridenko (2006)
Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation SchemesMath. Oper. Res., 31
-
K. Jansen, Lars Prädel (2014)
A New Asymptotic Approximation Algorithm for 3-Dimensional Strip Packing -
K. Jansen, M. Rau (2017)
Closing the gap for pseudo-polynomial strip packing -
Anna Adamaszek, T. Kociumaka, Marcin Pilipczuk, Michal Pilipczuk (2016)
Hardness of Approximation for Strip PackingACM Trans. Comput. Theory, 9
-
Joseph Mitchell (2021)
Approximating Maximum Independent Set for Rectangles in the Plane2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
-
J. Leung, T. Tam, C. Wong, G. Young, F. Chin (1990)
Packing Squares into a SquareJ. Parallel Distributed Comput., 10
-
claim holds for a subset of rectangles V (cid:3) of width at most δ · W such of vertical slices defined by V (cid:3) , namely V (cid:3) v slice , can be packed into the boxes
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
- ISSN
- 0178-4617
- eISSN
- 1432-0541
- DOI
- 10.1007/s00453-023-01130-2
- Publisher site
- See Article on Publisher Site
Abstract
In the Strip Packing problem, we are given a vertical half-strip [0,W]×[0,+∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[0,W]\times [0,+\infty )$$\end{document} and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. It is NP-hard to approximate this problem within a factor (3/2-ε)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(3/2-\varepsilon )$$\end{document} for any constant ε>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varepsilon >0$$\end{document} by a simple reduction from the Partition problem, while the current best approximation factor for it is (5/3+ε)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(5/3+\varepsilon )$$\end{document}. It seems plausible that Strip Packing admits a (3/2+ε)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(3/2+\varepsilon )$$\end{document}-approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta >0$$\end{document}, we call large the rectangles with width at least δW\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta W$$\end{document} and height at least δOPT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta OPT$$\end{document}, and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, the skewed case is still NP-hard to approximate within a factor (3/2-ε)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(3/2-\varepsilon )$$\end{document}, and we provide an (almost) tight (3/2+ε)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(3/2+\varepsilon )$$\end{document}-approximation algorithm.
Journal
Algorithmica – Springer Journals
Published: Oct 1, 2023
Keywords: Strip packing; Rectangle packing; Approximation algorithms