Given a mapped environment, we formulate the problem of visually tracking and following an evader using a probabilistic framework. In this work, we consider a non-holonomic robot with a limited visibility depth sensor in an indoor environment with obstacles. The mobile robot that follows the target is considered a pursuer and the agent being followed is considered an evader. We propose a probabilistic framework for both the pursuer and evader to achieve their conflicting goals. We introduce a smart evader that has information about the location of the pursuer. The goal of this variant of the evader is to avoid being tracked by the pursuer by using the visibility region information obtained from the pursuer, to further challenge the proposed smart pursuer. To validate the efficiency of the framework, we conduct several experiments in simulation by using Gazebo and evaluate the success rate of tracking an evader in various environments with different pursuer to evader speed ratios. Through our experiments we validate our hypothesis that a smart pursuer tracks an evader more effectively than a pursuer that just navigates in the environment randomly. We also validate that an evader that is aware of the actions of the pursuer is more successful at avoiding getting tracked by a smart pursuer than a random evader. Finally, we empirically show that while a smart pursuer does increase it's average success rate of tracking compared to a random pursuer, there is an increased variance in its success rate distribution when the evader becomes aware of its actions.
Megiddo @cite_12 proved that the complexity of searching a graph, which is equivalent to the search number, is NP-hard. In both these cases, visibility is not considered. The evader is found or captured if and only if both pursuer and evader are on the same vertices of the graph. The cop number is defined as a theoretical limit on the number of pursuers that can be used on a particular graph to successfully capture an evader. The cop number varies for different graphs based on the structure (topology) of the graph. Since we use a single pursuer in our work, we need not consider the cop number, instead we focus on finding a goal location for the pursuer that will maximize it's tracking rate.
@cite_12
2035417355 2154788235 2046635105 2053218689
We consider the game of Cops and Robbers played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to a new parameter, the capture time of a graph. While the capture time of a cop-win graph on n vertices is bounded above by n-3, half the number of vertices is sufficient for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have capture time within a small additive constant of the number of vertices. We consider the ratio of the capture time to the number of vertices, and extend this notion of capture time density to infinite graphs. For the infinite random graph, the capture time density can be any real number in [0,1]. We also consider the capture time when more than one cop is required to win. While the capture time can be calculated by a polynomial algorithm if the number k of cops is fixed, it is NP-complete to decide whether k cops can capture the robber in no more than t moves for every fixed t. T. Parsons originally proposed and studied the following pursuit-evasion problem on graphs: Members of a team of searchers traverse the edges of a graph G in pursuit of a fugitive, who moves along the edges of the graph with complete knowledge of the locations of the pursuers. What is the smallest number s ( G ) of searchers that will suffice for guaranteeing capture of the fugitive? It is shown that determining whether s ( G ) ≤ K , for a given integer K , is NP-complete for general graphs but can be solved in linear time for trees. We also provide a structural characterization of those graphs G with s ( G ) ≤ K for K = 1, 2, 3. Let G be a finite connected graph. Two players, called cop C and robber R, play a game on G according to the following rules. First C then R occupy some vertex of G. After that they move alternately along edges of G. The cop C wins if he succeeds in putting himself on top of the robber R, otherwise R wins. We review an algorithmic characterization and structural description due to Nowakowski and Winkler. Then we consider the general situation where n cops chase the robber. It is shown that there are graphs on which arbitrarily many cops are needed to catch the robber. In contrast to this result, we prove that for planar graphs 3 cops always suffice to win. In the graph searching game the opponents are a set of searchers and a fugitive in a graph. The searchers try to capture the fugitive by applying some sequence of moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive. In this paper, we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean (i.e. non-accessible to the fugitive) remain connected. We give evidence that many of the standard mathematical tools used so far in classic graph searching fail under the connectivity requirement. We also settle the question on ''the price of connectivity'', that is, how many searchers more are required for searching a graph when the connectivity demand is imposed. We make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees. In particular, for an n-vertex graph the ratio between the connected searching number and the non-connected one is O(logn) while for trees this ratio is always at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for all graphs. Our combinatorial results imply a complete characterization of connected graph searching on trees. It is based on a forbidden-graph characterization of the connected search number. We prove that the connected search game is monotone for trees, i.e. restricting search strategies to only those where the clean territories increase monotonically does not require more searchers. A consequence of our results is that the connected search number can be computed in polynomial time on trees, moreover, we show how to make this algorithm distributed. Finally, we reveal connections of this parameter to other invariants on trees such as the Horton-Strahler number.