Treasure evacuation with one robot on a disk

A robot is located at a point in the plane. A treasure and an exit, both stationary, are located at unknown (to the robot) positions both at distance one from the robot. Starting from its initial position, the robot aims to fetch the treasure to the exit. At any time the robot can move anywhere on the disk with constant speed. The robot detects an interesting point (treasure or exit) only if it passes over the exact location of that point. Given that an adversary controls the locations of both the treasure and the exit on the perimeter, we are interested in designing algorithms that minimize the treasure-evacuation time, i.e. the time it takes for the treasure to be discovered and brought to the exit by the robot. This seemingly simple treasure evacuation problem turns out to be surprisingly challenging. In this paper we differentiate how the robot's knowledge of the distance between the two interesting points affects the overall evacuation time. We demonstrate the difference between knowing only a lower bound of the distance versus knowing its the exact value, and provide search algorithms for both cases. In the former case we provide an algorithm which is shown to be optimal. In the latter case we give an algorithm which is off from the optimal algorithm (that does not know the locations of the treasure and the exit) by no more than 4 2 + 3 π + 2 6 2 + 2 π + 2 ≤ 1.019 multiplicatively, or π 2 − 2 ≤ 0.157 additively.