Domain-decomposition (DD) for Integral Equation can be achieved by aggregating standard basis functions into specialized basis functions on each sub-domain; this results in a strong compression of the MoM matrix, which allows an iteration-free (e.g., LU decomposition) solution also for electrically large problems. Fast matrix-vector product algorithms can be used in the matrix filling and compression process of the employed aggregate-functions approach: this hybrid approach has received considerable attention in recent literature. In order to quantitatively assess the performance, advantages and limitations of this class of methods, we start by proposing and demonstrating the use of the Adaptive Integral Method (AIM) fast factorization to accelerate the Synthetic Function eXpansion (SFX) DD approach. The method remains iteration free, with a significant boost in memory and time performances, with analytical predictions of complexity scalings confirmed by numerical results. Then, we address the complexity scaling of both stand-alone DD and its combined use with fast MoM; this is done analytically and discussed with respect to known literature accounts of various implementations of the DD paradigm, with nonobvious results that highlight needs and limitations, and yielding practical indications.