Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons

Alt, Helmut; Berg, Mark de; Knauer, Christian

doi:10.1007/978-3-662-48350-3_3

Cited by 7 publications

(10 citation statements)

References 13 publications

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“…This reduction increases the approximation ratio by a factor by 2. The reduction does not work when the pieces can be only translated, but Alt, de Berg, and Knauer [4] gave a 17.45-approximation algorithm for this problem using different techniques.…”

Section: Related Workmentioning

confidence: 99%

Online Packing to Minimize Area or Perimeter

Abrahamsen,

Beretta

2021

Preprint

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We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90 • or translations only.For the perimeter version we give algorithms with an absolute competitive ratio slightly less than 4 when only translations are allowed and when rotations are also allowed.We then turn our attention to minimizing the area and show that the asymptotic competitive ratio of any algorithm is at least Ω( √ n), where n is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases and the competitive ratio is thus optimal to within a constant factor. We also show that the competitive ratio cannot be bounded as a function of Opt. We then consider two special cases.The first is when all the given rectangles have aspect ratios bounded by some constant. The particular variant where all the rectangles are squares and we want to minimize the area of the bounding square has been studied before and an algorithm with a competitive ratio of 8 has been given [Fekete and Hoffmann, Algorithmica, 2017]. We improve the analysis of the algorithm and show that the ratio is at most 6, which is tight.The second special case is when all edges have length at least 1. Here, the Ω( √ n) lower bound still holds, and we turn our attention to lower bounds depending on Opt. We show that any algorithm for the translational case has an asymptotic competitive ratio of at least Ω( √ Opt). If rotations are allowed, we show a lower bound of Ω( 4 √ Opt). For both versions, we give algorithms that match the respective lower bounds: With translations only, this is just the algorithm from the general case with competitive ratio O( √ n) = O( √ Opt). If rotations are allowed, we give an algorithm with competitive ratio O(min{ √ n, 4 √ Opt}), thus matching both lower bounds simultaneously.

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Section: Related Workmentioning

confidence: 99%

Online Packing to Minimize Area or Perimeter

Abrahamsen,

Beretta

2021

Preprint

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“…Various shapes of objects were studied. In particular, cutting and packing problems for ellipses are presented in [3][4][5][6], circular packing problems are investigated in [7][8][9][10][11][12][13], and packing problems for convex polygons are considered in [11][12][13][14][15][16]. Papers [17][18][19][20][21] are devoted to irregular packing involving arbitrary shaped objects.…”

Section: Introductionmentioning

confidence: 99%

Optimized Packing Clusters of Objects in a Rectangular Container

Romanova

Панкратов

Litvinchev

et al. 2019

Mathematical Problems in Engineering

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A packing (layout) problem for a number of clusters (groups) composed of convex objects (e.g., circles, ellipses, or convex polygons) is considered. The clusters have to be packed into a given rectangular container subject to nonoverlapping between objects within a cluster. Each cluster is represented by the convex hull of objects that form the cluster. Two clusters are said to be nonoverlapping if their convex hulls do not overlap. A cluster is said to be entirely in the container if so is its convex hull. All objects in the cluster have the same shape (different sizes are allowed) and can be continuously translated and rotated. The objective of optimized packing is constructing a maximum sparse layout for clusters subject to nonoverlapping and containment conditions for clusters and objects. Here the term sparse means that clusters are sufficiently distant one from another. New quasi-phi-functions and phi-functions to describe analytically nonoverlapping, containment and distance constraints for clusters are introduced. The layout problem is then formulated as a nonlinear nonconvex continuous problem. A novel algorithm to search for locally optimal solutions is developed. Computational results are provided to demonstrate the efficiency of our approach. This research is motivated by a container-loading problem; however similar problems arise naturally in many other packing/cutting/clustering issues.

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“…Constant-factor approximation algorithms of polynomial running time have been found for many variants of packing problems, in particular for finding minimum-size rectangular or convex containers for a set of convex polygons under translations [2], that is, the objects may be translated but rotations are not allowed. Also, approximation algorithms for rigid motions (translations and rotations) are known in this case, see for instance [10].…”

Section: Introductionmentioning

confidence: 99%

Packing 2D Disks into a 3D Container

Alt

Cheong

Park

et al. 2018

WALCOM: Algorithms and Computation

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In this article, we consider the problem of finding in three dimensions a minimum-volume axisparallel box into which a given set of unit-radius disks can be packed under translations. The problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm based on reduction to finding a shortest Hamiltonian path in a weighted graph. As a byproduct, we can show that there is no finite size container into which all unit disks can be packed simultaneously.

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Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons

Cited by 7 publications

References 13 publications

Online Packing to Minimize Area or Perimeter

Online Packing to Minimize Area or Perimeter

Optimized Packing Clusters of Objects in a Rectangular Container

Packing 2D Disks into a 3D Container

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