Are there any 'tricks' to speed up sampling of a very large knapsack combination type prob?

Question

UPDATE: I have realized the problem below is not possible to answer in its current form because of the large amount of data involved(15k+ items). I just found out, the

Answer 1

I may be wrong but could this be viewed as a integer partition (was Triangle number, but I remembered I misremembered ) problem?

If that's the case, each entry would have a membership to a set of results for a given sum. Precalculating and caching the membership results for a given sum (yes a huge table) could form a very quick solution.

I could probably do it, but I'd need a large data set. Interesting though...

Consult the guru Knuth http://cs.utsa.edu/~wagner/knuth/fasc3b.pdf