I work in applied math. The bulk of my work is aimed at doing something efficiently. Given a problem (a physical model, plus some discretization), I'll have to find suitable algorithms to solve the resulting equations. I usually understand enough about the physics (and artificial physics introduced via the discrete setting) to have a complete understanding of the limits of the efficiency of any implementation. Nonetheless, actually creating the implementation which maximizes efficiency is not straightforward (otherwise I'd have no job!).

I guess you could classify this as two different problems: the base problem of understanding, in an abstract sense, the bounds on efficiency, and the actual problem of constructing algorithms which will touch those bounds. In my mind though, they feel intimately related in a way that I can't unravel.