Complete characterization of a class of privacy-preserving tracking problems
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We examine the problem of target tracking whilst simultaneously preserving the targets privacy as epitomized by the robotic panda-tracking scenario, which OKane introduced at the 2008 Workshop on the Algorithmic Foundations of Robotics to elegantly illustrate the utility of ignorance. The present paper reconsiders his formulation and the tracking strategy he proposed, along with its completeness. We explore how the capabilities of the robot and panda affect the feasibility of tracking with a privacy stipulation, uncovering intrinsic limits, no matter the strategy employed. This paper begins with a one-dimensional setting and, putting the trivially infeasible problems aside, analyzes the strategy space as a function of problem parameters. We show that it is not possible to actively track the target as well as protect its privacy for every non-trivial pair of tracking and privacy stipulations. Secondly, feasibility can be sensitive, in several cases, to the information available to the robot initially. Quite naturally in the one-dimensional model, one may quantify sensing power by the number of perceptual (or output) classes available to the robot. The robots power to achieve privacy-preserving tracking is bounded, converging asymptotically with increasing sensing power. We analyze the entire space of possible tracking problems, characterizing every instance as either achievable, constructively by giving a policy where one exists (some of which depend on the initial information), or proving that the instance is impossible. Finally, to relate some of the impossibility results in one dimension to their higher-dimensional counterparts, including the planar panda-tracking problem studied by OKane, we establish a connection between tracking dimensionality and the sensing power of a one-dimensional robot.
