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Pursuit-evasion

Pursuit-evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically (see e.g. Isaacs 1965). In 1976, Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit-evasion, and the graph formulation discrete pursuit-evasion (also called graph searching). Current research is typically limited to one of these two formulations.

Discrete formulation

In the discrete formulation of the pursuit-evasion problem, the environment is modeled as a graph.

Problem definition

There are innumerable possible variants of pursuit-evasion, though they tend to share many elements. A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a graph. The two sides take alternate turns, which consist of each member either staying put or moving along an edge to an adjacent node. If a pursuer occupies the same node as an evader the evader is captured and removed from the graph. The question usually posed is how many pursuers are necessary to ensure the eventual capture of all the evaders.

Often the movement rules are altered by changing the velocity of the evaders. This velocity is the maximum number of edges that an evader can move along in a single turn. In the example above, the evaders have a velocity of one. At the other extreme is the concept of infinite velocity, which allows an evader to move to any node in the graph so long as there is a path between its original and final positions that contains no nodes occupied by a pursuer. Similarly some variants arm the pursuers with "helicopters" which allow them to move to any vertex on their turn.

Other variants ignore the restriction that pursuers and evaders must always occupy a node and allow for the possibility that they are positioned somewhere along an edge. These variants are often referred to as sweeping problems, whilst the previous variants would fall under the category of searching problems.

Variants

Several variants are equivalent to important graph parameters. Specifically, finding the number of pursuers necessary to capture a single evader with infinite velocity in a graph G (when pursuers and evader are not constrained to move turn by turn, but move simultaneously) is equivalent to finding the treewidth of G. If this evader is invisible to the pursuers then the problem is equivalent to finding the pathwidth or vertex separation. Finding the number of pursuers necessary to capture a single invisible evader in a graph G in a single turn (that is, one movement by the pursuers from their initial deployment) is equivalent to finding the size of the minimum dominating set of G, assuming the pursuers can initially deploy wherever they like (this later assumption holds when pursuers and evader are assumed to move turn by turn).

The board game Scotland Yard is a variant of the pursuit-evasion problem.

Multi-Player Pursuit-Evasion Games

Solving multi-player pursuit-evasion games has also received increased attention. See R Vidal et al., Chung and Furukawa , Hespanha et al. and the references therein.

Continuous formulation

In the continuous formulation of pursuit-evasion games, the environment is modeled geometrically, typically taking the form of the Euclidean plane or another manifold. Variants of the game may impose maneuverability constraints on the players, such as a limited range of speed or acceleration. Obstacles may also be used.

Applications

One of the initial applications of the pursuit-evasion problem was missile guidance systems (Isaacs 1965).

References

• Ellis, J.; Sudborough, I.; Turner, J. (1994). "The vertex separation and search number of a graph". Information and Computation 113 (1): 50–79.
• Isaacs, R. (1965). Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. New York: John Wiley & Sons.
• Kiriousis, M.; Papadimitriou, C. (1986). "Searching and pebbling". Theoretical Computer Science 42 (2): 205–218.
• Nowakowski, R.; Winkler, P. (1983). "Vertex-to-vertex pursuit in a graph". Discrete Mathematics 43 235–239.
• Parsons, T. D. (1976). "Pursuit-evasion in a graph". Theory and Applications of Graphs, 426–441. .
• Seymour, P.; Thomas, R. (1993). "Graph searching, and a min-max theorem for tree-width". Journal of Combinatorial Theory, Series B 58 (1): 22–33.
• Vidal et al.; (2002). "Probabilistic pursuit-evasion games: theory, implementation, and experimental evaluation". IEEE Transactions on Robotics and Automation 18 (5): 662–669.
• Chung and Furukawa; "A Reachability-Based Strategy for the Time-Optimal Control of Autonomous Pursuers". Engineering Optimization To Appear .
• Hespanha et al. (1999). "Multiple-agent probabilistic pursuit-evasion games". Proceedings of the 38th IEEE Conference on Decision and Control., 2432-2437. .
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