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Finding minimal convex nested polygons

1989
*
Information and Computation
*

Our main result is

doi:10.1016/0890-5401(89)90049-7
fatcat:mxnxiazclbfflbqd4u2eldn6au
*an*O(n log k)*algorithm**for*solving*the**problem*, where n is*the*total number of vertices of*the*given*polygons*, and k is*the*number of vertices of a*minimal**nested**polygon*. ... We consider*the**problem*of finding a*polygon**nested*between two given convex*polygons*that has a*minimal*number of vertices. ...*The**problem*of determining a minimum*vertex**nested**polygon*belongs to*the*general area of*polygonal**approximations*. ...##
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STOCK CUTTING OF COMPLICATED DESIGNS BY COMPUTING MINIMAL NESTED POLYGONS

1995
*
Engineering optimization (Print)
*

With these results,

doi:10.1080/03052159508941261
fatcat:v2jjfae4ofcp5jc5opm432monq
*an*efficient*approximate**algorithm*is obtained*for**the*origina1 stock cutting*problem*. ... Numerica1 examples are provided to illustrate*the*working of*the**algorithm*in different cases. Key Words: stock cutting, design*approximation*, minima1*nested**polygon*,*approximate**algorithm*. ...*The*second author was supported, in part, by*The*Morris Hite Center at*The*University of Texas at Dallas. This support is gratefuIly acknowledged. ...##
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Page 443 of Mathematical Reviews Vol. , Issue 91A
[page]

1991
*
Mathematical Reviews
*

E. (6-IIS-F)

*An**approximate**algorithm**for**the**minimal**vertex**nested**polygon**problem*. Inform. Process. Lett. 33 (1989), no. 1, 35-44. ... Summary: “Given two simple*polygons*,*the**minimal**vertex**nested**polygon**problem*is*the**problem*of finding a*polygon**nested*be- tween*the*given*polygons*and having*the*minimum number of vertices. ...##
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IDENTIFYING ALTERNATE OPTIMAL SOLUTIONS TO THE DESIGN APPROXIMATION PROBLEM IN STOCK CUTTING

1999
*
Engineering optimization (Print)
*

Although there are

doi:10.1080/03052159908941378
fatcat:hqrxpyvbendrjb7j7wxblaekma
*algorithms*available to solve this*problem*, they all suffer from*an*undesirable feature that they only produce one optimal solution to*the**problem*, and do not identify*the*complete set ...*The*design*approximation**problem*is a well known*problem*in stock cutting, where, in order to facilitate*the*optimization techniques used in*the*cutting process, it is required to*approximate*complex designs ...*An*extended abstract of this paper appeared in*the*proceedings of*the*Canadian Conference on Computation Geometry held at Carleton University, Ottawa, in August 1996. ...##
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Efficient nesting of congruent convex figures

1984
*
Communications of the ACM
*

Optimal

doi:10.1145/357994.358022
fatcat:26lr37lpajhffpljbqiynupz4i
*nesting*is*the*arrangement of twodimensional*polygons*within a rectangular board so that waste is*minimized*. ... We first show that*the*original*problem*can be decomposed into two subproblems--one consisting of finding all convex paver*polygons*and*the*other of optimal (*minimal*waste) circumscription of*the*original ... SOLUTION OF*THE**NESTING**PROBLEM**Algorithms*3 and 4 provide*the*procedures*for**an*optimal circumscription of circular and basic hexagons by a parallel paver, respectively. ...##
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Separation and approximation of polyhedral objects

1995
*
Computational geometry
*

Given a family of disjoint

doi:10.1016/0925-7721(95)00006-u
fatcat:wzf7ohyjerdmvjsxrdwkblb2ny
*polygons*P" P" . . . , Pk in*the*plane, and*an*integer parameter m, it is AT-complete to decide if*the*Pi's can be pairwise separated by a*polygonal*family with at most m edges ... In three dimensions,*the**problem*is NP-complete even*for*two . 0925-7721/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO925-7721(95)00006-2 96 ... [ll give*an*O(n log n)*algorithm**for*finding a minimum-*vertex**polygon*separating two*nested*convex*polygons*.*An**algorithm*of similar complexity is given by Wang and Chan 95-114 97 Theorem 5.1. ...##
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Very Fast Approximation of the Matrix Chain Product Problem

1996
*
Journal of Algorithms
*

This paper considers

doi:10.1006/jagm.1996.0037
fatcat:nbo2umellratzfzwr2gssyzpce
*the*matrix chain product*problem*. This*problem*can be Ž . solved in O n log n sequential time, while*the*best known parallel NC*algorithm*Ž 2 . ... This paper presents a very fast parallel*algorithm**for**approximately*solving*the*matrix chain product*problem*and*for**the**problem**for*Ž . finding a near-optimal triangulation of a convex*polygon*. ... Since¨)¨) max¨,¨*for*each p i -p -r or r -p -j ,*APPROXIMATING**THE*MATRIX CHAIN PRODUCT*PROBLEM*In this section we show that*the**approximation**algorithm*presented in*the*previous section can be applied ...##
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Optimal positioning of irregular shapes in stamping die strip

2010
*
The International Journal of Advanced Manufacturing Technology
*

Given two nonconvex

doi:10.1007/s00170-010-2772-6
fatcat:oi4smb5cerfefhfvaq7vonvev4
*polygons*,*the**algorithm*is able to calculate their NFPh very quickly and without any*approximation*by a*polygon*clipping method. ...*The**nesting*of two-dimensional shapes is a common*problem*, where raw material has to be economically cut. ... Acknowledgments This work was performed with*the*financial support of*the*Italian Ministry of University and Research. ...##
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An optimal polygonal boundary encoding scheme in the rate distortion sense

1998
*
IEEE Transactions on Image Processing
*

We

doi:10.1109/83.650847
pmid:18267376
fatcat:mafqxobstngondhyxmbajxi2pa
*approximate**the*boundary by a*polygon*, and consider*the**problem*of finding*the**polygon*which leads to*the*smallest distortion*for*a given number of bits. ... We also address*the*dual*problem*of finding*the**polygon*which leads to*the*smallest bit rate*for*a given distortion. We consider two different classes of distortion measures. ...*The*key observation*for*deriving*an*efficient search*for**the**polygon*that*minimizes**the*unconstrained*problem*(16) is based on*the*fact that given a certain*vertex*of a*polygon*and*the*Lagrangian cost ...##
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Covering Polygons with Rectangles
[chapter]

2017
*
Lecture Notes in Computer Science
*

A well-known and well-investigated family of hard optimization

doi:10.1007/978-3-319-55911-7_20
fatcat:ga3jmpqeqvg6dho55tj4iopgh4
*problems*concerns variants of*the*cutting stock or*nesting**problem*, i.e.*the*non-overlapping placing of*polygons*to be cut from a rectangle ... or*the*plane whilst*minimizing**the*waste. ... Acknowledgments*The*author is grateful to Torben Hagerup, Christian Rähtz, Lev Sorokin and*the*anonymous reviewers*for*valuable hints and remarks. ...##
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A simple probablistic algorithm for approximating two and three-dimensional objects

1999
*
Canadian Conference on Computational Geometry
*

5 Acknowledgements I thank Cao

dblp:conf/cccg/Zhu99
fatcat:x2lkh4br3vcy3fldjc6xyhpzxq
*An*Wang, D.T. Lee*for*comments in*the*early stage of this work. ... Computing such*an*optimal minimum size polyhedron O is NP-complete. We present*the*following*algorithm*to compute*an**approximate*solution*for*O . ... ., a pair of*nested*convex*polygons*P;Q with Q P, we want t o*approximate**the*minimum size convex*polygon*contained in P , Q. ...##
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Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards
[article]

2018
*
arXiv
*
pre-print

We present here three polynomial-time

arXiv:1712.05492v2
fatcat:5mzfhao6rbd23agylg4sjr4qme
*algorithms*with a constant*approximation*ratio*for*guarding*an*n-sided simple*polygon*P using*vertex*guards. ... Most standard versions of this*problem*are known to be NP-hard. In 1987, Ghosh provided a deterministic O( n)-*approximation**algorithm**for**the*case of*vertex*guards and edge guards in simple*polygons*. ...*for**the**problem*with*an**approximation*ratio better than ((1 − )/12) ln n*for*any > 0, unless NP = P. ...##
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Computing Nonsimple Polygons of Minimum Perimeter
[article]

2016
*
arXiv
*
pre-print

We provide exact and

arXiv:1603.07077v1
fatcat:fejltaj42fc6tiqwh5pbfthfry
*approximation*methods*for*solving a geometric relaxation of*the*Traveling Salesman*Problem*(TSP) that occurs in curve reconstruction:*for*a given set of vertices in*the*plane,*the*...*problem*Minimum Perimeter*Polygon*(MPP) asks*for*a (not necessarily simply connected)*polygon*with shortest possible boundary length. ... Irina Kostitsyna is supported by*the*Netherlands Organisation*for*Scientific Research (NWO) under project no. 639.023.208. ...##
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Graph-Theoretic Solutions to Computational Geometry Problems
[article]

2009
*
arXiv
*
pre-print

Many

arXiv:0908.3916v1
fatcat:s4kizimp7jggliyivcqm5iokmm
*problems*in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing*an*auxiliary graph and performing a graph-theoretic*algorithm*on it. ... Often,*the*efficiency of*the**algorithm*depends on*the*special properties of*the*graph constructed in this way. ... Acknowledgements This work was supported in part by NSF grant 0830403 and by*the*Office of Naval Research under grant N00014-08-1-1015. ...##
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Active-set sequential quadratic programming method with compact neighbourhood algorithm for the multi-polygon mass production cutting-stock problem with rotatable polygons

2009
*
International Journal of Production Economics
*

This study presents

doi:10.1016/j.ijpe.2009.01.014
fatcat:uyzjcmyerbhz5ejf74uep3vaka
*an*overlap index and it is much more suitable*for**the*active-set SQP method which can reduce*the*time spend*for*constraint consideration. ...*problem*. ... Luo,*an*associate professor of National United University,*for*his suggestions on this study. ...
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