Exact optimal cost for priority evacuation from a disk (two wireless agents)

Determine the exact optimal worst-case evacuation cost for the priority evacuation problem on the unit disk with two unit-speed agents in the wireless communication model, i.e., the weighted group search problem on the disk with weight parameter w=0 where the objective is the termination time of the designated agent (the queen). Concretely, establish the minimum possible value of the objective over all feasible trajectories, thereby closing the gap between the best known lower and upper bounds for this setting.

Background

The paper studies weighted group search on the unit disk with two unit-speed agents operating in the wireless model. The cost function is a weighted average g_w(x,y)=wx+y of the agents’ termination times; the case w=0 coincides with the priority evacuation problem, in which termination is measured solely by the time the designated agent (queen) reaches the target.

Prior to this work, the best known bounds for priority evacuation on the disk were an upper bound of 4.81854 and a lower bound of 4.38962. This paper introduces a new lower-bound framework based on linear programming relaxations and improves the lower bound to 4.56798, reducing but not eliminating the gap. Determining the exact optimal value (and thus closing the remaining gap) remains unresolved.