# A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis

The problem of evacuating two robots from the disk in the face-to-face model was first introduced by Czyzowicz et al. [DISC’2014], and has been extensively studied (along with many variations) ever since with respect to worst-case analysis. We initiate the study of the same problem with respect to average-case analysis, which is also equivalent to designing randomized algorithms for the problem. In particular, we introduce constrained optimization problem _{2}Evac_{𝐹2𝐹}, in which one is trying to minimize the average-case cost of the evacuation algorithm given that the worst-case cost does not exceed *w*. The problem is of special interest with respect to practical applications, since a common objective in search-and-rescue operations is to minimize the average completion time, given that a certain worst-case threshold is not exceeded, e.g., for safety or limited energy reasons. Our main contribution is the design and analysis of families of new evacuation parameterized algorithms which can solve _{2}Evac_{𝐹2𝐹}, for every *w* for which the problem is feasible. Notably, the worst-case analysis of the problem, since its introduction, has been relying on technical numerical, computer-assisted calculations, following tedious robot trajectory analysis. Part of our contribution is a novel systematic procedure, which given *any evacuation algorithm*, can derive its worst- and average-case performance in a clean and unified way.