Is subset sum with a fixed target sum NP-complete? I've read that subset sum is NP-complete. What happens when I change the decision problem to look for a  constant number? So the decision problem would look like this:


  Input:
  A collection of nonnegative integers A and a nonnegative integer b, 
  
  Output:
  Boolean value indicating whether some subset of 
  the collection sums to 10


Would this still be NP-complete? I don't believe you would be able to reduce every other NP-complete problem to it.
We can throw away all zeroes in $A$. If a subset of $A$ sums to $10$, then this subset contains at most $10$ elements. There are at most $10|A|^{10}$ such non-empty subsets, so we can go over all of them in polynomial time.