I have some vague recollections from my algorithms courses, but nothing specific comes to mind.

let S be a multiset of things we are interested in.

As input, we get T, s₁, s₂, … s_n, all of which are subsets of S, except that T can have duplicates .

As output we get whether it is possible to pick exactly 1 element from each s_i so that we end up with T. What would be really nice is if in the case of failure it told you which s_i‘s were not covered by T (although this isn’t really well defined when there’s non-empty intersections in the s_i‘s).

In my case, most of the time there won’t be many elements which appear in more than one s_i‘s, but I have to account for the possibility. Also, an ‘online’ algorithm in which you provide the s_i‘s up front, and then pass in elements of T one at a time would be convenient.

This is almost like the exact hitting set problem, but I am given T, rather than trying to find a subset of T which has this property.