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### Covering convex sets with non-overlapping polygons

Herbert Edelsbrunner, Arch D. Robison, Xiao-Jun Shen
1990 Discrete Mathematics
We prove that given n 2 3 convex, compact, and pairwise disjoint sets in the plane, they may be covered with n non-overlapping convex polygons with a total of not more than 6n -9 sides, and with not more  ...  Furthermore, we construct sets that require 6n -9 sides and 3n -6 slopes for n 2 3. The upper bound on the number of slopes implies a new bound on a recently studied transversal problem.  ...  Covering convex sets with non-overlapping polygons Maximal polygons and their contact graph. Fig. 7 . 7 Avoiding overlapping edges.  ...

### Rotational polygon overlap minimization and compaction

Victor J. Milenkovic
1998 Computational geometry
Overlap minimization is modified to create a practical algorithm for compaction: starting with a non-overlapping layout in a rectangular container, plan a non-overlapping motion that diminishes the length  ...  An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P 1 , P 2 , P 3 , . . . , P k in a container polygon Q, translate and rotate the polygons  ...  Inner convex covers (ICC) of non-convex apparel polygon with 71 vertices. "Fat" ICC (left) has 39 polygons but covers each point 24 times on average.  ...

### Covering points with orthogonally convex polygons

Burkay Genç, Cem Evrendilek, Brahim Hnich
2011 Computational geometry
In this paper, we address the problem of covering points with orthogonally convex polygons.  ...  In particular, given a point set of size n on the plane, we aim at finding if there exists an orthogonally convex polygon such that each edge of the polygon covers exactly one point and each point is covered  ...  Counting all solutions So far we have provided a method for obtaining a covering of a point set with an orthogonally convex polygon, if such a covering exists.  ...

### An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee [chapter]

Stephan Eidenbenz, Peter Widmayer
2001 Lecture Notes in Computer Science
The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP -hard, even for polygons without holes  .  ...  As a second step, we use dynamic programming to obtain a convex polygon which is maximum with respect to the number of "basic triangles" that are not yet covered by another convex polygon.  ...  Introduction and Problem Definition The problem Minimum Convex Cover is the problem of covering a given polygon T with a minimum number of (possibly overlapping) convex polygons that lie in T .  ...

### An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee

Stephan J. Eidenbenz, Peter Widmayer
2003 SIAM journal on computing (Print)
The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP -hard, even for polygons without holes  .  ...  As a second step, we use dynamic programming to obtain a convex polygon which is maximum with respect to the number of "basic triangles" that are not yet covered by another convex polygon.  ...  Introduction and Problem Definition The problem Minimum Convex Cover is the problem of covering a given polygon T with a minimum number of (possibly overlapping) convex polygons that lie in T .  ...

### Optimal space coverage with white convex polygons

2014 Journal of combinatorial optimization
The goal is to find a set of convex polygons with maximum total area that cover all white points and exclude all black points.  ...  We study the problem on three different settings (based on overlapping between different convex polygons): (1) In case convex polygons are permitted to have common area, we present a polynomial algorithm  ...  . 1a ) -Non-overlapping convex covering (NOCC) In this problem WCPs are not allowed to have common area but are allowed to have common vertices (Fig. 1b) .  ...

### INNER-COVER OF NON-CONVEX SHAPES

DANIEL COHEN-OR, SHULY LEV-YEHUDI, ADI KAROL, AYELLET TAL
2003 International journal of shape modeling
We present an algorithm that for a given simple non-convex polygon P finds an approximate inner-cover by large convex polygons.  ...  The algorithm then builds a set of large convex polygons contained in P by constructing the convex hulls of subsets of C.  ...  PRELIMINARIES Let P be a simple polygon. We seek to generate a set of possibly overlapping, convex polygons that satisfy a few requirements described below.  ...

### Finite packing and covering by congruent convex domains

K�roly B�r�czky
2003 Discrete & Computational Geometry
Finally, let K be centrally symmetric, and let D n be the convex domain with minimal area containing n non-overlapping congruent copies of K .  ...  For a convex domain K , let H (K ) be a circumscribed polygon with at most six sides whose area is minimal, and letH (K ) be an inscribed hexagon with at most six sides whose area is maximal.  ...  There exists a covering 1 , . . . , n of D by convex, compact sets such that each i ⊂ K i , the sets 1 , . . . , n are pairwise non-crossing, and i A( i ) is minimal under the previous two conditions.  ...

### An efficient algorithm for the regular W1 packing of polygons in the infinite plane

P D Watson, A M Tobias
1999 Journal of the Operational Research Society
Polygons F—N inclusive were chosen as a set of typical simple non-convex polygons having no more than four notches.  ...  Much of this early work was concerned with the mathematical properties of convex polygons which cover the plane without gaps (tilings). It is well known?  ...

### Computing Covers of Plane Forests [article]

Luis Barba, Alexis Beingessner, Prosenjit Bose, Michiel H. M. Smid
2013 arXiv   pre-print
A ϕ-cover of a set T={T_1, T_2, ..., T_m} of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that each tree T_i is contained in some region of the cover, and  ...  We present two properties for the function ϕ that make the ϕ-cover well-defined. Examples for such functions ϕ are the convex hull and the axis-aligned bounding box.  ...  Let a weakly disjoint pair of convex polygons P , Q be a pair of convex polygons such that P \ Q and Q \ P are both connected sets of points, and P does not share a vertex with Q.  ...

### Approximating the Maximum Overlap of Polygons under Translation [article]

Sariel Har-Peled, Subhro Roy
2014 arXiv   pre-print
We present an (1-ε)-approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P.  ...  Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts.  ...  For a non-convex polygon P with n vertices and r notches, Keil and Snoeyink [KS02] solves the minimal convex decomposition problem in O(n + r 2 min(r 2 , n)) time, that is, they compute a decomposition  ...

### Covering polygons is hard

J.C. Culberson, R.A. Reckhow
1988 [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
with convex polygons, covering an orthogonal polygon with rectangles, and 4. covering the boundary of an orthogonal polygon with rectangles.  ...  We show that the following minimum cover problems are NP-hard, even for polygons without holes: 1. covering an arbitrary polygon with conveg polygons, 2. covering the boundary of .an arbitrary polygon  ...  Franzblau In contrmt, the problem of decomposing polygons into a minimum number of non-overlapping convex polygons has a pdlynomial time solution for polygons without holes [4,3,10,13].  ...

### Page 5626 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews
For a class Q of planar sets a convex set W is called a covering set if any member of Q can be covered by a congruent copy of W W is called a minimal covering set if it cannot be replaced by a proper subset  ...  For a (planar) convex body K, it is defined as the minimum number of mutually non-overlapping congruent copies such that they can touch K and prevent any other congruent copy from touching K without overlapping  ...

### A new mixed-integer programming model for irregular strip packing based on vertical slices with a reproducible survey [article]

Juan J. Lastra-Díaz, M. Teresa Ortuño
2022 arXiv   pre-print
The irregular strip-packing problem, also known as nesting or marker making, is defined as the automatic computation of a non-overlapping placement of a set of non-convex polygons onto a rectangular strip  ...  In order to improve the efficiency of the current MIP models, this work introduces a new family of continuous MIP models based on a novel formulation of the NoFit-Polygon Covering Model (NFP-CM), called  ...  Appendix B: The reproducible experiments on irregular strip-packing This appendix introduces a detailed reproducibility protocol and dataset  providing our raw output data together with a collection  ...

### Compaction and separation algorithms for non-convex polygons and their applications

Zhenyu Li, Victor Milenkovic
1995 European Journal of Operational Research
Given a two dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied \forces."  ...  We also consider the problem of separating overlapping polygons using a minimal amount of motion and show it to be NP-complete.  ...  of the set of non-overlapping positions of each pair of polygons.  ...
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