Linear algorithm to find minimum subset sum over a threshold

Question

I have a collection of N positive integers, each bounded by a (relatively small) constant C. I want to find a subset of these numbers with the smallest sum greater than (or

Answer 1

Well one solution is to:

T = {0}

for x in V
   for t in T
       T.insert(x+t)

for i in K to max(T)
   if (T.contains(i))
       return i

fail

This gives you the size of the subset, but you can adapt to output the members.

The maximum size of T is O(N) (because of C bound), so the running time is O(N^2) and the space is O(N). You can use a bit array of length NC as the backing store of T.