Exact wake-up ratio in the Euclidean plane

Determine whether the maximum makespan (wake-up ratio) for the Freeze-Tag Problem in the Euclidean plane with the ℓ2 norm equals 1 + 2√2, by proving or disproving that the configuration with four robots at (0,1), (0,−1), (1,0), and (−1,0) attains the global maximum makespan among all finite instances with all sleeping robots within unit distance of the initial active robot.

Background

The paper studies the wake-up ratio γ{d,p} for the Euclidean Freeze-Tag Problem and improves the best-known upper bound for γ{2,2} from 4.62 to 4.31, while a lower bound around 3.83 is known via a specific four-robot configuration.

Prior work has conjectured that this particular four-robot configuration is worst-case for (ℝ^2, ℓ2), which would imply γ{2,2} = 1 + 2√2 ≈ 3.83. Confirming or refuting this conjecture would settle the exact value of γ{2,2}.