The work analyzes the computational-complexity savings that a single bit of feedback can provide in the computationally intense setting of non-ergodic MIMO communications. Specifically we derive upper bounds on the feedback-aided complexity exponent required for the broad families of ML-based and lattice based decoders to achieve the optimal diversity-multiplexing behavior. The bounds reveal a complexity that is reduced from being exponential in the number of codeword bits, to being at most exponential in the rate. Finally the derived savings are met by practically constructed ARQ schemes, as well as simple lattice designs, decoders and computation-halting policies. {\textcopyright} 2012 IEEE.