Proceedings ArticleDOI
Covering orthogonal polygons with star polygons: the perfect graph approach
Rajeev Motwani,Arvind Raghunathan,Huzur Saran +2 more
- pp 211-223
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TLDR
In the case where the polygon has at most three dent orientations, thePolygon covering problem can be reduced to the problem of covering a triangulated (chordal) graph with a minimum number of cliques.Abstract:
We consider the problem of covering simple orthogonal polygons with star polygons. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q.In general, orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques. We thus obtain a O (n10) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship.In the case where the polygon has at most three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a triangulated (chordal) graph with a minimum number of cliques. This gives us an O (n3) algorithm.read more
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Proceedings ArticleDOI
Covering polygons is hard
Joseph Culberson,R. A. Reckhow +1 more
TL;DR: 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.
Book ChapterDOI
On the rectilinear art gallery problem
TL;DR: In this article, it was shown that [n/4] point guards are always sufficient and sometimes necessary to watch a rectilinear polygon with an arbitrary number of holes, where n is the total number of vertices.
Journal ArticleDOI
Guarding orthogonal art galleries with sliding cameras
Matthew J. Katz,Gila Morgenstern +1 more
TL;DR: In this paper, the authors study the problem of guarding an orthogonal art gallery with security cameras sliding back and forth along straight tracks, and they show that if only vertical (alternatively, horizontal) tracks are allowed, then a solution minimizing the number of tracks can be found in polynomial time, and if both orientations are allowed and both tracks have different orientations, then they give a 2-approximation for x-monotone galleries.
Proceedings Article
On the Rectilinear Art Gallery Problem (Extended Abstract)
TL;DR: It is shown that [n/4] point guards are always sufficient and sometimes necessary to watch a rectilinear polygon with an arbitrary number of holes, where n is the total number of vertices.
References
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Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
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Algorithmic graph theory and perfect graphs
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
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Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
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The ellipsoid method and its consequences in combinatorial optimization
TL;DR: The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.