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Searching Polyhedra by Rotating HalfPlanes
, 2012
"... 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 halfplanes of illumination. ..."
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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 halfplanes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar oneway sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no welldefined 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 NPhard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NPhard, and that deciding if a given target area is searchable at all is strongly PSPACEhard, 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. 1 1
The 3dimensional Searchlight Scheduling Problem
"... The problem of searching for a mobile intruder in a polygonal region by a set of stationary guards, each carrying an orientable laser, is known in the literature as the Searchlight Scheduling Problem. A longstanding conjecture concerns the NPhardness of deciding if a given polygon is searchable by ..."
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The problem of searching for a mobile intruder in a polygonal region by a set of stationary guards, each carrying an orientable laser, is known in the literature as the Searchlight Scheduling Problem. A longstanding conjecture concerns the NPhardness of deciding if a given polygon is searchable by a given set of guards. In this paper we introduce the more general problem of detecting an intruder in a 3dimensional polyhedral region by a set of searchplanes within a given time, and we prove its NPhardness.
Monitoring the Plane with Rotating Radars
"... Consider a set P of n points in the plane and n radars located at these points. The radars are rotating perpetually (around their centre) with identical constant speeds, continuously emitting pulses of radio waves (modelled as halfinfinite rays). A radar can “locate ” (or detect) any object in the ..."
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Consider a set P of n points in the plane and n radars located at these points. The radars are rotating perpetually (around their centre) with identical constant speeds, continuously emitting pulses of radio waves (modelled as halfinfinite rays). A radar can “locate ” (or detect) any object in the plane (e.g., using radio echolocation when its ray is incident to the object). We propose a model for monitoring the plane based on a system of radars. For any point p in the plane, we define the idle time of p, as the maximum time that p is “unattended ” by any of the radars. We study the following monitoring problem: What should the initial direction of the n radar rays be so as to minimize the maximum idle time of any point in the plane? We propose algorithms for specifying the initial directions of the radar rays and prove bounds on the idle time depending on the type of configuration of n points. For arbitrary sets P we give a O(n logn) time algorithm guaranteeing a O(1/ n) upper bound on the idle time, and a O(n6 / ln3 n) time algorithm with associated O(logn/n) upper bound on the idle time. For a convex set P, we show a O(n logn) time algorithm with associated O(1/n) upper bound on the idle time. Further, for any set P of points if the radar rays are assigned a direction independently at random with the uniform distribution then we can prove a tight Θ(lnn/n) upper and lower bound on the idle time with high probability.
Distributed Optimization for Radar Mission Coordination
"... Abstract — This paper presents optimization algorithms that enable multiple shipbased radar systems to maximize their collective target search area while concurrently solving an optimal sensortotarget assignment problem. We present theoretically justified strategies that determine the optimal sh ..."
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Abstract — This paper presents optimization algorithms that enable multiple shipbased radar systems to maximize their collective target search area while concurrently solving an optimal sensortotarget assignment problem. We present theoretically justified strategies that determine the optimal ship location and searcharea radius for each radar in a given threat environment satisfying a predefined resource reserve constraint. We then solve the optimal targetassignment problem by balancing the radar tasking among all participating sensors and show the resulting increase in the number of trackable targets in the combined search area. We provide analytical and numerical simulations to illustrate the algorithm performance. I.
Partial Searchlight Scheduling is Strongly PSPACEcomplete
"... The problem of searching a polygonal region for an unpredictably moving intruder by a set of stationary guards, each carrying an orientable laser, is known as the Searchlight Scheduling Problem. Determining the complexity of deciding if the entire area can be searched is a longstanding open problem ..."
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The problem of searching a polygonal region for an unpredictably moving intruder by a set of stationary guards, each carrying an orientable laser, is known as the Searchlight Scheduling Problem. Determining the complexity of deciding if the entire area can be searched is a longstanding open problem. Recently, the author introduced the Partial Searchlight Scheduling Problem, in which only a given subregion of the environment has to be searched, and proved that its 3dimensional decision version is PSPACEhard, even when restricted to orthogonal polyhedra. Here we extend and refine this result, by proving that 2dimensional Partial Searchlight Scheduling is strongly PSPACEcomplete, both in general and restricted to orthogonal polygons in which the region to be searched is a rectangle.
DISTRIBUTED OPTIMIZATION OF RESOURCE ALLOCATION FOR SEARCH AND TRACK ASSIGNMENT WITH MULTIFUNCTION RADARS
, 2013
"... The longterm goal of this research is to contribute to the design of a conceptual architecture and framework for the distributed coordination of multifunction radar systems. The specific research objective of this dissertation is to apply results from graph theory, probabilistic optimization, and ..."
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The longterm goal of this research is to contribute to the design of a conceptual architecture and framework for the distributed coordination of multifunction radar systems. The specific research objective of this dissertation is to apply results from graph theory, probabilistic optimization, and consensus control to the problem of distributed optimization of resource allocation for multifunction radars coordinating on their search and track assignments. For multiple radars communicating on a radar network, cooperation and agreement on a network resource management strategy increases the group’s collective search and track capability as compared to noncooperative radars. Existing resource management approaches for a single multifunction radar optimize the radar’s configuration by modifying the radar waveform and beampattern. Also, multiradar approaches implement a topdown, centralized sensor management framework that relies on fused sensor data, which may be impractical due to bandwidth constraints. This dissertation presents a distributed radar resource optimization approach for a network of multifunction radars. Linear and nonlinear models estimate the