Proceedings ArticleDOI
Covering polygons is hard
Joseph Culberson,R. A. Reckhow +1 more
- Vol. 17, Iss: 1, pp 601-611
Reads0
Chats0
TLDR
It is shown that the following minimum cover problems are NP-hard, even for polygons without holes, and these results hold even if the polygons are required to be in general position.Abstract:
It is shown that the following minimum cover problems are NP-hard, even for polygons without holes: (1) covering an arbitrary polygon with convex polygons; (2) covering the boundary of an arbitrary polygon with convex polygons; (3) covering an orthogonal polygon with rectangles; and (4) covering the boundary of an orthogonal polygon with rectangles. It is noted that these results hold even if the polygons are required to be in general position. >read more
Citations
More filters
Book ChapterDOI
Art Gallery and Illumination Problems
TL;DR: Since Victor Klee's question, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others.
Recent Results in Art Galleries
TL;DR: The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point of P is visible from some point of G.
Journal ArticleDOI
Recent results in art galleries (geometry)
TL;DR: The author provides an introduction to art gallery theorems and surveys the recent results of the field, examining several new problems that have the same geometric flavor as art gallery problems.
Journal ArticleDOI
Inapproximability Results for Guarding Polygons and Terrains
TL;DR: This paper proves that if the input polygon has no holes, there is a constant δ >0 such that no polynomial time algorithm can achieve an approximation ratio of 1+δ, for each of these guard problems, and shows inapproximability for the POINT GUARD problem for polygons with holes.
Journal ArticleDOI
On the difficulty of triangulating three-dimensional nonconvex polyhedra.
Jim Ruppert,Raimund Seidel +1 more
TL;DR: The problem of deciding whether a given polyhedron can be triangulated is NP-complete, and hence likely to be computationally intractable.
References
More filters
Journal ArticleDOI
Some NP-hard polygon decomposition problems
Joseph O'Rourke,K. Supowit +1 more
TL;DR: Three polygon decomposition problems are shown to be NP-hard and thus unlikely to admit efficient algorithms, and the polygonal region is permitted to contain holes.
Journal ArticleDOI
Decomposing a Polygon into Simpler Components
TL;DR: This paper considers decompositions which do not introduce Steiner points, and applies a technique for improving the efficiency of dynamic programming algorithms to achieve polynomial time algorithms for the problems of decomposing a simple polygon into the minimum number of each of the component types.
Book ChapterDOI
The Power of Non-Rectilinear Holes
TL;DR: Four multiconnected-polygon partition problems are shown to be NP-hard.
Proceedings ArticleDOI
Decomposing a polygon into its convex parts
Bernard Chazelle,David P. Dobkin +1 more
TL;DR: This work considers the problem of decomposing a simple (non-convex) polygon into the union of a minimal number of convex polygons and reaches polynomial time bounded algorithms for exact solution as well as low degree polynometric time bounded algorithm/or approximation methods.
Journal ArticleDOI
Covering Regions by Rectangles
TL;DR: In this article, the authors prove a conjecture of Chvatal that if a board is convex in the horizontal and vertical directions, then the minimum number of rectangles whose union is a rectilinear subset of the board is the maximum cardinality of an antirectangle.