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The Parameterized Complexity of Guarding Almost Convex Polygons

Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi, Danny Z. Chen, Sergio Cabello
2020 International Symposium on Computational Geometry  
To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art [...]  ...  The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point  ...  Acknowledgements We thank anonymous reviewers for helpful comments that improved and simplified the paper.  ... 
doi:10.4230/lipics.socg.2020.3 dblp:conf/compgeom/AgrawalKL0Z20 fatcat:ojkrkrplhfgt5gaigushtkt77q

The Parameterized Complexity of Guarding Almost Convex Polygons [article]

Akanksha Agrawal, Kristine V.K. Knudsen, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi
2020 arXiv   pre-print
We utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in  ...  The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point  ...  Acknowledgements We thank anonymous reviewers for helpful comments that improved and simplified the paper.  ... 
arXiv:2003.07793v1 fatcat:agnxduu3jfaa3gpv2u3ksjsxji

Page 553 of Mathematical Reviews Vol. , Issue 90A [page]

1990 Mathematical Reviews  
We also provide an application of our approximate algorithm to guarding art galleries with almost the minimum number of watchmen.”  ...  Summary: “This note investigates the computational complexity of a special kind of decomposition of polygons into convex, star- shaped, and spiral components.  ... 

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem [article]

Simon Hengeveld, Tillmann Miltzow
2022 arXiv   pre-print
A polygon P has vision stability δ if the optimal number of enhanced guards to guard P is the same as the optimal number of diminished guards to guard P.  ...  Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.  ...  Special thanks goes to Matthew Drescher and Emile Palmieri-Adant for attempting to implement previous versions of the algorithm.  ... 
arXiv:2007.06920v2 fatcat:pjp677voyja3zgebydp5bpvrie

The Parameterized Hardness of the Art Gallery Problem [article]

Édouard Bonnet, Tillmann Miltzow
2017 arXiv   pre-print
The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard.  ...  These lower bounds almost match the n^O(k) algorithms that exist for both problems.  ...  Despite the recent successes of parameterized complexity, only very few results on the art gallery problem are known.  ... 
arXiv:1603.08116v2 fatcat:ou2xlkjpmvbljgaj6jfyt6rmw4

A Flexible Framework for Covering and Partitioning Problems in Indoor Spaces

Sung-Hwan Kim, Ki-Joune Li, Hwan-Gue Cho
2020 ISPRS International Journal of Geo-Information  
One of the main features of our framework is the parameterized constraint, which characterizes the properties and restrictions of unit geometries used for the covering and partitioning tasks formulated  ...  We apply it to particular applications, device placement and route planning problems, in order to give examples of the use of our framework in the perspective on how to design a constraint and how to use  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/ijgi9110618 fatcat:knzvfzosjncqrpt5tbfrmpdtgy

Topological Art in Simple Galleries [article]

Daniel Bertschinger, Nicolas El Maalouly, Tillmann Miltzow, Patrick Schnider, Simon Weber
2021 arXiv   pre-print
Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P.  ...  We say two points a,b∈ P can see each other if the line segment seg(a,b) is contained in P. We denote by V(P) the family of all minimum guard placements.  ...  The Parameterized Complexity of Guarding Almost Convex Polygons. In SoCG 2020, LIPIcs, pages 3:1–3:16, 2020. [5] Akanksha Agrawal and Meirav Zehavi.  ... 
arXiv:2108.04007v1 fatcat:qifivyhmgrbzrn4fridq2iw2aq

Face-guarding polyhedra

Giovanni Viglietta
2014 Computational geometry  
The problem can be informally stated as: how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate it?  ...  Along the way we discuss some applications, arguing that face guards are not a reasonable model for guards patrolling on the surface of a polyhedron.  ...  Essentially, there are cases in which the path that such a patrolling guard ought to follow is so complex (in terms of the number of turns, if it is a polygonal chain) that a much simpler path, striving  ... 
doi:10.1016/j.comgeo.2014.04.009 fatcat:h3dhgq4wqngtplfmg47fbsafiq

Parameterized Hardness of Art Gallery Problems

Édouard Bonnet, Tillmann Miltzow
2020 ACM Transactions on Algorithms  
of algorithms → Parameterized complexity and exact algorithms. 10  ...  The Vertex Guard Art Gallery problem asks for 4 such a set S subset of the vertices of P. A point in the set S is referred to as a guard.  ...  We build a sub-polygon that can be seen 535 entirely by pairs of convex vertices if and only if they correspond to the same 2-element. 536 For each j ∈ [k], permutation σ j will be encoded by a sub-polygon  ... 
doi:10.1145/3398684 fatcat:25gzjq2u2bbkhi3pte4wqj5awy

Face-Guarding Polyhedra [article]

Giovanni Viglietta
2014 arXiv   pre-print
The problem can be informally stated as: how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate its interior?  ...  We give upper and lower bounds on the minimum number of faces required to guard the interior of a given polyhedron in each of these classes, in terms of the total number of its faces, f.  ...  Acknowledgments The author wishes to thank the anonymous reviewers for precious suggestions on how to improve the readability of this paper.  ... 
arXiv:1305.2866v4 fatcat:ihb7recnfnd7lgnrhppaqo3tdy

An Approximation Algorithm for the Art Gallery Problem

Édouard Bonnet, Tillmann Miltzow, Marc Herbstritt
2017 International Symposium on Computational Geometry  
Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem.  ...  The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards.  ...  The main application of his paper is to yield an approximation algorithm for a variant of the point guard art gallery problem when one is allowed to guard only almost all the polygon.  ... 
doi:10.4230/lipics.socg.2017.20 dblp:conf/compgeom/BonnetM17 fatcat:xceaikxucngtjmhf4ioflhkrty

Orthogonal Terrain Guarding is NP-complete

Édouard Bonnet, Panos Giannopoulos, Marc Herbstritt
2018 International Symposium on Computational Geometry  
They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or  ...  Terrain Guarding can be seen as a special case of the famous art gallery problem where one has to place at most k guards on a terrain made of n vertices in order to fully see it.  ...  The principal remaining open questions concern the parameterized complexity of terrain guarding. (1) Is Terrain Guarding FPT parameterized by the number of guards?  ... 
doi:10.4230/lipics.socg.2018.11 dblp:conf/compgeom/BonnetG18 fatcat:eslflf4nivechlpn3odmpdse34

OBBTree

S. Gottschalk, M. C. Lin, D. Manocha
1996 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques - SIGGRAPH '96  
In particular, it can robustly and accurately detect all the contacts between large complex geometries composed of hundreds of thousands of polygons at interactive rates.  ...  We present a data structure and an algorithm for e cient and exact interference detection amongst complex models undergoing rigid motion. The algorithm is applicable to all general polygonal models.  ...  Model Complexity: The input models are composed of many h undreds of thousands of polygons. 2.  ... 
doi:10.1145/237170.237244 dblp:conf/siggraph/GottschalkLM96 fatcat:4u2ie2bdazaipepynkptcmbsvi

Orthogonal Terrain Guarding is NP-complete [article]

Édouard Bonnet, Panos Giannopoulos
2018 arXiv   pre-print
They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or  ...  Terrain Guarding can be seen as a special case of the famous art gallery problem where one has to place at most $k$ guards on a terrain made of $n$ vertices in order to fully see it.  ...  This contrasts with the parameterized complexity of the more general art gallery problem where an algorithm running in time f (k)n o(k/ log k) for any computable function f would disprove the ETH, both  ... 
arXiv:1710.00386v2 fatcat:db5sahext5hj3iei5vuvdz72u4

Pseudo-Triangulations - a Survey [article]

Guenter Rote, Francisco Santos, Ileana Streinu
2007 arXiv   pre-print
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles.  ...  Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces.  ...  We thank the referees for their extensive comments.  ... 
arXiv:math/0612672v2 fatcat:adhyppd3wjbhnfmdky5dbxjxzu
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