3,159 research outputs found
Patrolling a Street Network is Strongly NP-Complete but in P for Tree Structures
We consider the following problem: Given a finite set of straight line
segments in the plane, determine the positions of a minimal number of points on
the segments, from which guards can see all segments. This problem can be
interpreted as looking for a minimal number of locations of policemen, guards,
cameras or other sensors, that can observe a network of streets, corridors,
tunnels, tubes, etc. We show that the problem is strongly NP-complete even for
a set of segments with a cubic graph structure, but in P for tree structures
A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras
Consider a sliding camera that travels back and forth along an orthogonal
line segment inside an orthogonal polygon with vertices. The camera
can see a point inside if and only if there exists a line segment
containing that crosses at a right angle and is completely contained in
. In the minimum sliding cameras (MSC) problem, the objective is to guard
with the minimum number of sliding cameras. In this paper, we give an
-time -approximation algorithm to the MSC problem on any
simple orthogonal polygon with vertices, answering a question posed by Katz
and Morgenstern (2011). To the best of our knowledge, this is the first
constant-factor approximation algorithm for this problem.Comment: 11 page
Searching Polyhedra by Rotating Half-Planes
The Searchlight Scheduling Problem was first studied in 2D polygons, where
the goal is for point guards in fixed positions to rotate searchlights to catch
an evasive intruder. Here the problem is extended to 3D polyhedra, with the
guards now boundary segments who rotate half-planes of illumination. After
carefully detailing the 3D model, several results are established. The first is
a nearly direct extension of the planar one-way sweep strategy using what we
call exhaustive guards, a generalization that succeeds despite there being no
well-defined notion in 3D of planar "clockwise rotation". Next follow two
results: every polyhedron with r>0 reflex edges can be searched by at most r^2
suitably placed guards, whereas just r guards suffice if the polyhedron is
orthogonal. (Minimizing the number of guards to search a given polyhedron is
easily seen to be NP-hard.) Finally we show that deciding whether a given set
of guards has a successful search schedule is strongly NP-hard, and that
deciding if a given target area is searchable at all is strongly PSPACE-hard,
even for orthogonal polyhedra. A number of peripheral results are proved en
route to these central theorems, and several open problems remain for future
work.Comment: 45 pages, 26 figure
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