Escaping Off-Line Searchers and a Discrete Isoperimetric Theorem

We study the following pursuit-evasion game: Imagine a hunting trip on a foggy day. The field is an n x n square grid where the prey, a rabbit, is restricted to this field. The hunters and the rabbit move alternately, and each group can choose whether to stay at its current position or to move to an adjacent grid. Each hunter can see only his own grid, meaning that hunter A can not know hunter B’s location unless they planned their movements before the hunt began. How many hunters would be needed to guarantee their prize? IntroductIon The pursuit-evasion game has many variations depending on the domain of the players’ positions, the information available to the players, the relative speed of the pursuers and the evader, and the definition of capture. The problem is addressed in the robotics community for its applications in collision avoidance, search and rescue, and air-traffic control. The terms pursuer-evader, hunter-rabbit, and searcher-target are used synonymously. In our paper, we study the model of offline searchers, which was introduced by Dumitrescu et al. The search path for each searcher is determined beforehand, and the question is whether the target can avoid capture using the information of those given search paths. Given the domain of the grid, it is easy to imagine an inescapable row of n searchers sweeping the square grid, but if the number of searchers decreases, the target may escape. Dumistrescu et al. showed that O(√n) offline searchers cannot capture the target in the n x n grid. We improve this bound. rESultS Our main result is a recursive characterization of the model and a discrete isoperimetric theorem for the grid graph. We show that +1 searchers suffice on the square grid and that the target evades searchers. In Figure 1, Red Circles represent current locations of searchers and Green Circles represent locations of searchers in the previous stage. A cross inside the green circle represents the location(vertice) in which all its neighbors are forbidden on the last stage or forbidden due to searchers on current stage. A cross without a circle is a forbidden vertex that accumulated two stages before the current stage. Theorem 1 searchers are never enough Define Forb(a) as the set of forbidden vertices p in V which a target cannot inhabit at stage a, and si(a) is the location of the i th searcher at that stage. Sa is the set of all searchers at stage a. [ ] 2 Stage 1 Stage 2 to 3 Stage 1 to 2 Stage 2 [ ] 2 figure 1: illustration of the events that take in the grid as the searchers advance. [ ] 2