Memory needed for computational graph
Your problem sounds similar to one-shot (black) pebbling. Wu, Austrin, Pitassi, and Liu, in their paper titled Inapproximability of treewidth, one-shot pebbling, and related layout problems (J. Artificial Intelligence Res. 49 (2014), 569–600), show that it is (probably) hard to compute the optimal cost (which corresponds to your memory).