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### Finding minimal convex nested polygons

Alok Aggarwal, Heather Booth, Joseph O'Rourke, Subhash Suri, Chee K. Yap
1989 Information and Computation
Our main result is 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.  ...

### STOCK CUTTING OF COMPLICATED DESIGNS BY COMPUTING MINIMAL NESTED POLYGONS

1995 Engineering optimization (Print)
With these results, 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.  ...

### 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.  ...

### IDENTIFYING ALTERNATE OPTIMAL SOLUTIONS TO THE DESIGN APPROXIMATION PROBLEM IN STOCK CUTTING

1999 Engineering optimization (Print)
Although there are 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.  ...

### Efficient nesting of congruent convex figures

Dov Dori, Moshe Ben-Bassat
1984 Communications of the ACM
Optimal 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.  ...

### Separation and approximation of polyhedral objects

Joseph S.B. Mitchell, Subhash Suri
1995 Computational geometry
Given a family of disjoint 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.  ...

### Very Fast Approximation of the Matrix Chain Product Problem

Artur Czumaj
1996 Journal of Algorithms
This paper considers 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  ...

### Optimal positioning of irregular shapes in stamping die strip

Roberto Licari, E. Lo Valvo
2010 The International Journal of Advanced Manufacturing Technology
Given two nonconvex 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.  ...

### An optimal polygonal boundary encoding scheme in the rate distortion sense

G.M. Schuster, A.K. Katsaggelos
1998 IEEE Transactions on Image Processing
We 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  ...

### Covering Polygons with Rectangles [chapter]

Roland Glück
2017 Lecture Notes in Computer Science
A well-known and well-investigated family of hard optimization 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.  ...

### A simple probablistic algorithm for approximating two and three-dimensional objects

Binhai Zhu
1999 Canadian Conference on Computational Geometry
5 Acknowledgements I thank Cao 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.  ...

### Constant Approximation Algorithms for Guarding Simple Polygons using Vertex Guards [article]

Pritam Bhattacharya, Subir Kumar Ghosh, Sudebkumar Pal
2018 arXiv   pre-print
We present here three polynomial-time 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.  ...

### Computing Nonsimple Polygons of Minimum Perimeter [article]

Sándor P. Fekete, Andreas Haas, Michael Hemmer, Michael Hoffmann, Irina Kostitsyna, Dominik Krupke, Florian Maurer, Joseph S. B. Mitchell, Arne Schmidt, Christiane Schmidt, Julian Troegel
2016 arXiv   pre-print
We provide exact and 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.  ...

### Graph-Theoretic Solutions to Computational Geometry Problems [article]

David Eppstein
2009 arXiv   pre-print
Many 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.  ...

### Active-set sequential quadratic programming method with compact neighbourhood algorithm for the multi-polygon mass production cutting-stock problem with rotatable polygons

M.T. Yu, T.Y. Lin, C. Hung
2009 International Journal of Production Economics
This study presents 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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