The participant dependencies make multiple ‘cores’ plus some number of tails around each core. From any group we must take at least the core and at most the entire group.

This is a knapsack problem where each item has variable size. Solve this using a modified version of the standard dynamic programming - make a table possible[N][K], which denotes whether it is possible to admit K participants using the first N groups. Fill it in with the following recurrence relation:

possible[N][K] = possible[N-1][K] or (possible[N-1][K-i] | min_size[N] <= i <= max_size[N])