Solving new problems can be made easier if one can build on experiences with other problems one has already successfully solved. The ability to exploit earlier problem-solving experiences in solving new problems seems to require several cognitive sub-abilities. Minimally, one needs to be able to retrieve relevant knowledge of earlier solved problems and their solutions (solved-exemplar retrieval), to determine whether or not a retrieved problem is sufficient analogous to the problem at hand (analogy derivation), and to infer how the solution-method used for the old problem can be used for the new problem (candidate inference projection). All three processes have successfully been modeled under the framework of Structure-Mapping Theory (SMT). It has long been known that analogy derivation under SMT is computationally intractable, meaning that all (exact) algorithms implementing this ability run impractically long. In this paper we show that the same holds for the other two sub-processes. In sharp contrast to this theoretical intractability, empirical research reveals that in certain situations humans can quickly retrieve appropriate problem-exemplars and quickly make goal-relevant candidate inference projections. How can this speed of processing be explained within the framework of SMT? We consider several possible explanations, both existing and new, and assess their explanatory validity by performing computational-level complexity analyses. Our analyses not only reveal that explanations that have been conjectured to date are incomplete but also identify a set of complete explanations that can guide future empirical research on analogical problem solving.