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

Unfortunately, there is little hope to find an algorithm which is much better than brute-force, considering that the problem is actually NP-hard (but not even NP-complete).

A proof of NP-hardness of this problem is that the minimum vertex cover problem (well known to be NP-hard and not NP-complete) is easily reducible to it:

Given a graph. Let's create package P_v for each vertex v of the graph. Also create package X what "and"-requires (P_u or P_v) for each edge (u, v) of the graph. Find a minimum set of packages to be installed in order to satisfy X. Then v is in the minimum vertex cover of the graph iff the corresponding package P_v is in the installation set.