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Search and pursuit-evasion in mobile robotics
- Autonomous Robots
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On Discrete-time Pursuit-evasion Games with Sensing Limitations
, 2007
"... We address discrete-time pursuit-evasion games in the plane where every player has identical sensing and motion ranges restricted to closed discs of given sensing and stepping radii. A single evader is initially located inside a bounded subset of the environment and does not move until detected. We ..."
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Cited by 10 (2 self)
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We address discrete-time pursuit-evasion games in the plane where every player has identical sensing and motion ranges restricted to closed discs of given sensing and stepping radii. A single evader is initially located inside a bounded subset of the environment and does not move until detected. We propose a Sweep-Pursuit-Capture pursuer strategy to capture the evader and apply it to two variants of the game: the first involves a single pursuer and an evader in a bounded convex environment and the second involves multiple pursuers and an evader in a boundaryless environment. In the first game, we give a sufficient condition on the ratio of sensing to stepping radius of the players that guarantees capture. In the second, we determine the minimum probability of capture, which is a function of a novel pursuer formation and independent of the initial evader location. The Sweep and Pursuit phases reduce both games to previously-studied problems with unlimited range sensing. Thereafter, we demonstrate how capture is achieved using available strategies. We obtain novel upper bounds on the capture time and present simulation studies that suggest robustness to sensing errors.
Variations on Cops and Robbers
"... We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R> 1 edges at a time, establishing a general upper bound of n/α (1−o(1))√logα n 1, where α = 1 + R, thus generalizing the best known upper bound for the classical case R = 1 due to ..."
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Cited by 10 (1 self)
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We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R> 1 edges at a time, establishing a general upper bound of n/α (1−o(1))√logα n 1, where α = 1 + R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of 1 1− an n-vertex graph can be as large as n R−2 for finite R, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is at most O(n(log log n) 2 / logn). Our approach is based on expansion. 1
Cop and Robber Games when the Robber can Hide and Ride
, 2011
"... In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G =(V,E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber ..."
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Cited by 9 (4 self)
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In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G =(V,E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski and Winkler [Discrete Math., 43 (1983), pp. 235–239] and Quilliot [Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes, Thèsededoctorat d’état, Université de Paris VI, Paris, 1983] characterized the cop-win graphs as graphs admitting a dismantling scheme. In this paper, we characterize in a similar way the class CWFR(s, s ′)ofcop-win graphs in the game in which the robber and the cop move at different speeds s and s ′ , s ′ ≤ s. We also establish some connections between cop-win graphs for this game with s ′ <sand Gromov’s hyperbolicity. In the particular case s =2ands ′ = 1, we prove that the class of cop-win graphs is exactly the well-known class of dually chordal graphs. We show that all classes CWFR(s, 1), s ≥ 3, coincide, and we provide a structural characterization of these graphs. We also investigate several dismantling schemes necessary or sufficient for the cop-win graphs in the game in which the robber is visible only every k moves for a fixed integer k>1. In particular, we characterize the graphs which are cop-win for any value of k.
The Role of Information in the Cop-Robber Game
, 2008
"... We investigate the role of the information available to the players on the outcome of the cops and robbers game. This game takes place on a graph and players move along the edges in turns. The cops win the game if they can move onto the robber’s vertex. In the standard formulation, it is assumed tha ..."
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Cited by 8 (1 self)
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We investigate the role of the information available to the players on the outcome of the cops and robbers game. This game takes place on a graph and players move along the edges in turns. The cops win the game if they can move onto the robber’s vertex. In the standard formulation, it is assumed that the players can “see ” each other at all times. A graph G is called cop-win if a single cop can capture the robber on G. We study the effect of reducing the cop’s visibility. On the positive side, with a simple argument, we show that a cop with small or no visibility can capture the robber on any cop-win graph (even if the robber still has global visibility). On the negative side, we show that the reduction in cop’s visibility can result in an exponential increase in the capture time. Finally, we start the investigation of the variant where the visibility powers of the two players are symmetric. We show that the cop can establish eye contact with the robber on any graph and present a sufficient condition for capture. In establishing this condition, we present a characterization of graphs on which a natural greedy pursuit strategy suffices for capturing the robber.
Sensing limitations in the Lion and Man problem
- IN ACC
, 2007
"... ... in a bounded, convex, planar environment in which both players have identical sensing ranges, restricted to closed discs about their current locations. The evader is randomly located inside the environment and moves only when detected. The players can step inside identical closed discs, centered ..."
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Cited by 7 (1 self)
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... in a bounded, convex, planar environment in which both players have identical sensing ranges, restricted to closed discs about their current locations. The evader is randomly located inside the environment and moves only when detected. The players can step inside identical closed discs, centered at their respective positions. We propose a sweep-pursuit-capture strategy for the pursuer to capture the evader. The sweep phase is a search operation by the pursuer to detect an evader within its sensing radius. In the pursuit phase, the pursuer employs a greedy strategy of moving towards the last-sensed evader position. We show that in finite time, the problem reduces to a previously-studied problem with unlimited sensing, which allows us to use the established Lion strategy in the capture phase. We give a novel upper bound on the time required for the pursuit phase to terminate using the greedy strategy and a sufficient condition for this strategy to work in terms of the value of the ratio of sensing to stepping radius of the players.
Probabilistic Graph-Clear
- In Proceedings of the IEEE International Conference on Robotics and Automation
, 2009
"... Abstract — This paper introduces a probabilistic model for multirobot surveillance applications with limited range and possibly faulty sensors. Sensors are described with a footprint and a false negative probability, i.e. the probability of failing to report a target within their sensing range. The ..."
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Cited by 6 (3 self)
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Abstract — This paper introduces a probabilistic model for multirobot surveillance applications with limited range and possibly faulty sensors. Sensors are described with a footprint and a false negative probability, i.e. the probability of failing to report a target within their sensing range. The model implements a probabilistic extension to our formerly developed deterministic approach for modeling surveillance tasks in large environments with large robot teams known as Graph-Clear. This extension leads to a new algorithm that allows to answer new design and performance questions, namely 1) how many robots are needed to obtain a certain confidence that the environment is free from intruders, and 2) given a certain number of robots, how should they coordinate their actions to minimize their failure rate. I.
Chasing a Fast Robber on Planar Graphs and Random Graphs
"... We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c∞(G) denote the number of cops needed to capture the robber in a graph G in this va ..."
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Cited by 5 (1 self)
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We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c∞(G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G. We show that if G is planar then c∞(G) = Θ(tw(G)), and there is a constant-factor approximation algorithm for computing c∞(G). We also determine, up to constant factors, the value of c∞(G) of the Erdős-Rényi random graph G = G(n, p) for all admissible values of p, and show that when the average degree is ω(1), c∞(G) is typically asymptotic to the domination number.
Complete information pursuit evasion in polygonal environments
- In Proc. of 25th Conference on Artificial Intelligence (AAAI
"... Suppose an unpredictable evader is free to move around in a polygonal environment of arbitrary complexity that is under full camera surveillance. How many pursuers, each with the same maximum speed as the evader, are necessary and sufficient to guarantee a successful capture of the evader? The pursu ..."
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Cited by 4 (2 self)
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Suppose an unpredictable evader is free to move around in a polygonal environment of arbitrary complexity that is under full camera surveillance. How many pursuers, each with the same maximum speed as the evader, are necessary and sufficient to guarantee a successful capture of the evader? The pursuers always know the evader’s current position through the camera network, but need to physically reach the evader to capture it. We allow the evader the knowledge of the current positions of all the pursuers as well—this accords with the standard worst-case analysis model, but also models a practical situation where the evader has “hacked ” into the surveillance system. Our main result is to prove that three pursuers are always sufficient and sometimes necessary to capture the evader. The bound is independent of the number of vertices or holes in the polygonal environment.
