what are the assumptions inherent in the social problems perspective on crime?
There are a few.
And books have been written. Have you read Crime and Punishment? It is a classic. http://en.wikipedia.org/wiki/Crime_and_Punishment
Entire college courses in criminology exist on the subject. Many sociologists examine the lives of criminals, and the factors that seem to commonly popup.
The biggest assumption is that there are "causes" of crime.