I want to show that a problem is NP-hard by reducing a known NP-complete problem to it. However, I will have to use a special case of the NP-complete problem for the reduction to work. I'm pretty sure that the special case version is also NP-complete but I have no idea how to prove that. Are there any general guidelines for how to do this?
For example, consider this version of the SUBSET SUM problem without repetition:
Given an integer I and a multiset S of integers in the range 1,2,...,10, is there a non-empty subset of S whose sum is I?
I could be wrong, but don't think the restriction of possible values in S to {1, ..., 10} affects the NP-completeness of the problem. How would one go about showing this?
EDIT: Apparently that version of the problem is actually in P. I might restate my question later.