It's an amazing fact that if the $C$ is a symmetric monoidal category so that its components form a group, then $NC$ is an infinite loop space. Now if we have a Waldhausen category $D$, a category with distinguished cofibrations and weak equivalences, we can also construct an infinite loop-space via the S - construction.
Are these two constructions equivalent? In other words, given such a $C$ can I construct a Waldhausen category so the S - construction gives something weakly equivalent to $NC$? And if $D$ is a Waldhausen category can I construct a symmetric monoidal category so its nerve is weakly equivalent to the S - construction?