The Role of Visibility in Pursuit/Evasion Games

The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The “classic” CR version is a perfect information game: the cops’ (pursuer’s) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit/evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops’ point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

The Role of Visibility in Pursuit/Evasion Games

The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The “classic” CR version is a perfect information game: the cops’ (pursuer’s) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit/evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops’ point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

Lower Bound of Zero-visibility Cops and Robber Game on the Cube Grid

Zero-visibility cops and robber game is a variant of the classical model. The robber is invisible in the zero-visibility game. In this paper, we study the zero-visibility cops and robber game on the cube grid. We first study a partition problem of the cube grid. Then we prove the lower bound on the zero-visibility cop number of the cube grid by using the results in the partition. We also show the lower bound is a quadratic polynomial about n.