Has anyone seen this problem before? It’s supposed to be NP-complete.
We are given vertices V_1,…,V_n and possible parent sets for each vertex. Each parent set has an associated cost. Let O be an ordering (a permutation) of the vertices. We say that a parent set of a vertex V_i is consistent with an ordering O if all of the parents come before the vertex in the ordering. Let mcc(V_i, O) be minimum cost of the parent sets of vertex V_i that are consistent with ordering O. I need to find an ordering O that minimizes the total cost: mcc(V_1, O), … ,mcc(V_n, O).
I don’t quite understand the part “…if all of the parents come before the vertex in the ordering.” What does it mean?
No, I haven’t seen that problem before.
As for the bit you’re not sure about – an ordering is just an order of all the vertices, so I think “if all the parents come before the vertex in the ordering” just means exactly what it says. For instance, say (A, B) is one parent set of D: that parent set is consistent with the ordering [A,B,C,D], since A and B are before D, and not consistent with the ordering [A,D,B,C], since B is after D; however, say (A) is another parent set of D – that one is consistent with both those orderings. Does that make sense?