Question

I am wondering whether this this is an established computer science problem and if there is any polynomial time solution or approximation

Suppose I have some list X made up of true and false values

X = [True, False, True, False, True...True]

I also have a set of other lists which are the same length as X, with true and false values

A = [False, True, True, True, True, False .... False]
B = [False, False, True, False, True, False .... False]
...etc

Now, I want to find a ‘sum’ of these other lists (which is applying the bitwise OR operator to each element.. i.e. F + F = F , F + T = T , T + T = T) that best explain the observations seen in list X (I can introduce a scoring system that gives some score for a match and a penalty for a mismatch in the solution), and since there could be many possible solutions, I want to impose a penalty to the algorithm for the more lists it uses in its solution.

Answer 1

The problem that you’re describing is NP-hard by a reduction from the minimum set cover problem, which is known to be NP-hard.

The reduction is as follows. Given a set S of n elements, create a list of n copies of “true” as your list X. Then, for each set that might be permitted in the set cover, replace it with a list that has true or false in each spot based on whether the set contains or does not contain the given element of S. If you assign a penalty of infinity to a mismatch and assign a cost of one to each list, then there is a set cover of size k or less in the original set cover problem if and only if your problem has a solution of cost k or less.

This means that there is no known polynomial time algorithm for this problem, and you will need to either accept approximate answers or be willing to let your program run for a long time on some inputs.

Hope this helps!