I did some readings on the subject, and learned that this is the classic Global Postage Stamp Problem, which in the general case is an unsolved problem in number theory and computationally intractable (which of course makes the year-long contest the more interesting).
Here are some papers on the subject:
- A Postage Stamp Problem (Alter & Barnett, 1980)
- Algorithms for Computing the h-Range of the Postage Stamp Problem (Mossige, 1981)
- A Postage Stamp Problem (Lunnon, 1986)
- Two New Techniques for Computing Extremal h-bases Ak (Challis, 1992)