Dependency Algorithm - find a minimum set of packages to install

Question

I\'m working on an algorithm which goal is to find a minimum set of packages to install package \"X\".

I\'ll explain better with an example:

Answer 1

To add to Misandrist's answer: your problem is NP-complete NP-hard (see dened's answer).

Edit: Here is a direct reduction of a Set Cover instance (U,S) to your "package problem" instance: make each point z of the ground set U an AND requirement for X. Make each set in S that covers a point z an OR requirement for z. Then the solution for package problem gives the minimum set cover.

Equivalently, you can ask which satisfying assignment of a monotone boolean circuit has fewest true variables, see these lecture notes.