Imagine we have a multiple-choice exam with N questions. Suppose we have a set of K answer sheets to the exam and their total scores (1 for a correct answer on a question, 0 for incorrect).
How much information can we extract from this set about the true answers to the exam, and what is the shape of the space of possible answers after extracting this information? (e.g., is it convex in some sense? is it a linear subspace? or something like that)
E.g. if we have at least one answer sheet with a score of N, then we know all the correct answers. If we have an answer sheet with a score of 0, then we can exclude at least one answer from each question. [If we have two different answer sheets both with a score of N-1, we know the correct answers to all questions in which they agree] - (on further thought, this is actually incorrect..). Etc.
What's this problem called? Is it solvable in polynomial time? (looks like it's reducible to integer programming)