An Optimal Algorithm for Online Freeze-tag

In the freeze-tag problem, one active robot must wake up many frozen robots. The robots are considered as points in a metric space, where active robots move at a constant rate and activate other robots by visiting them. In the (time-dependent) online variant of the problem, frozen robots are not revealed until a specified time. Hammar, Nilsson, and Persson have shown that no online algorithm can achieve a competitive ratio better than $7/3$ for online freeze-tag, and asked whether there is any $O(1)$-competitive algorithm. In this paper, we provide a $(1+\sqrt{2})$-competitive algorithm for online time-dependent freeze-tag, and show that no algorithm can achieve a lower competitive ratio on every metric space.