The memory dumping problem arises in the context of planning and scheduling activities of the Mars Express mission of the European Space Agency. The problem consists of scheduling scientific data transmission from Mars to the Earth. A previously developed algorithm computes robust schedules in a heuristic way and iteratively improves the schedule robustness by solving a sequence of max-flow problems. We present a linear programming algorithm to compute schedules of maximum robustness, providing provably optimal solutions in a very short computing time. We also give necessary and sufficient conditions to characterize "easy" and "difficult" instances, such that the former ones can be solved directly without any optimization algorithm.