181 research outputs found
On -Guarding Thin Orthogonal Polygons
Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point
to guard a point if and only if the minimum axis-aligned rectangle spanned
by and is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of -guards is polynomial for tree polygons, but the
run-time was . We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with holes or thickness , becoming fixed-parameter tractable in
.Comment: 18 page
Mobile vs. point guards
We study the problem of guarding orthogonal art galleries with horizontal
mobile guards (alternatively, vertical) and point guards, using "rectangular
vision". We prove a sharp bound on the minimum number of point guards required
to cover the gallery in terms of the minimum number of vertical mobile guards
and the minimum number of horizontal mobile guards required to cover the
gallery. Furthermore, we show that the latter two numbers can be calculated in
linear time.Comment: This version covers a previously missing case in both Phase 2 &
Guarding the vertices of thin orthogonal polygons is NP-hard
An orthogonal polygon of P is called “thin” if the dual graph of the partition obtained by extending all edges of P towards its interior until they hit the boundary is a tree. We show that the problem of computing a minimum guard set for either a thin orthogonal polygon or only its vertices is NP-hard, indeed APX-hard, either for guards lying on the boundary or on vertices of the polygon.Fondo Europeo de Desarrollo RegionalFundação para a Ciência e a Tecnologi
Securing Pathways with Orthogonal Robots
The protection of pathways holds immense significance across various domains,
including urban planning, transportation, surveillance, and security. This
article introduces a groundbreaking approach to safeguarding pathways by
employing orthogonal robots. The study specifically addresses the challenge of
efficiently guarding orthogonal areas with the minimum number of orthogonal
robots. The primary focus is on orthogonal pathways, characterized by a
path-like dual graph of vertical decomposition. It is demonstrated that
determining the minimum number of orthogonal robots for pathways can be
achieved in linear time. However, it is essential to note that the general
problem of finding the minimum number of robots for simple polygons with
general visibility, even in the orthogonal case, is known to be NP-hard.
Emphasis is placed on the flexibility of placing robots anywhere within the
polygon, whether on the boundary or in the interior.Comment: 8 pages, 5 figure
The Dispersive Art Gallery Problem
We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery problem we are given a polygon ? and a real number ?, and want to decide whether ? has a guard set such that every pair of guards in this set is at least a distance of ? apart.
In this paper, we study the vertex guard variant of this problem for the class of polyominoes. We consider rectangular visibility and distances as geodesics in the L?-metric. Our results are as follows. We give a (simple) thin polyomino such that every guard set has minimum pairwise distances of at most 3. On the positive side, we describe an algorithm that computes guard sets for simple polyominoes that match this upper bound, i.e., the algorithm constructs worst-case optimal solutions. We also study the computational complexity of computing guard sets that maximize the smallest distance between all pairs of guards within the guard sets. We prove that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP-complete.
We were also able to find an optimal dynamic programming approach that computes a guard set that maximizes the minimum pairwise distance between guards in tree-shaped polyominoes, i.e., computes optimal solutions; due to space constraints, details can be found in the full version of our paper [Christian Rieck and Christian Scheffer, 2022]. Because the shapes constructed in the NP-hardness reduction are thin as well (but have holes), this result completes the case for thin polyominoes
Guarding and Searching Polyhedra
Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings.
In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra.
We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time.
Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1.
We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards.
All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding.
Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes
around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra.
Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard.
Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known
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