Directed Acyclic Graph with multi-parent nodes - graph-algorithm

Question

Given: A directed acyclic graph with weighted edges, where a node can have multiple parents.
Problem: For each child of root node, find a minimum-cost(sum of weights) path from such a child to some leaf which can be reached. A node can only be present in one such min-cost paths.
Example graph:
In the above graph, for node 2, all the available paths are:
2 -> 5
2 -> 1 -> 9 -> 6
2 -> 1 -> 10 -> 6
Among which 2 -> 1 -> 10 -> 6 has minimum cost of 3.5
Similarly, for node 4, all the available paths are:
4 -> 11 -> 8
4 -> 7 -> 10 -> 6
Among which 4 -> 7 -> 10 -> 6 has minimum cost of 3.0
The result currently is:
For node 2, the path is 2 -> 1 -> 10 -> 6 : Cost 3.5
For node 4, the path is 4 -> 7 -> 10 -> 6 : Cost 3.0
For node 3, there is no such path.
I have written a code which does this. Now, if such min-cost paths didn't have any node in common, the algorithm would stop and give me min-cost paths for all children of root node.
However, if there exists a node in common, I have to retain it only in one of them. The reason is that normally such multi-parent nodes are due to noisy data. A node is supposed to belong to only one parent. I am trying to keep such a node in a path which has minimum cost. So here, node 10 belongs to min-path of node 4 which has cost of 3.0, compared to min-path of node 2 which has cost of 3.5. Same logic with node 6 too. So, I will just compare the costs to dis-associate some of the multi-parent nodes. Dis-association doesn't mean the edge will be removed. All I do is that save best parent for each node within node's data-structure. For example, Node 10 will have an entry saying "best parent is node 7" and Node 6 will have an entry "best parent is node 10". I can actually remove the edge itself but I may want the whole graph structure to remain intact for any future computations.
So, the logic looks like this:
Do:
For each child node of the root:
Find out min-cost path. Store that path and the cost.
If conflicting paths exist:
Compare the costs of conflicting paths and save "best parent" for each node.
While there were conflicting paths;
Questions:
Does this logic makes sense? I am worried that this iterative way of eliminating conflicts may not converge for some graphs. For example, while re-calculating the min-path for node 2, if 2 -> 5 is found to be the min-path now and assume if node 5 is being used in some other node's min-path during first iteration, then I would have to re-assign "best parent" for node 5 as node 2 and re-iterate. In a nut-shell, everytime I try to fix some node's min-path, I may change the other's. Can such an algorithm converge to some solution? If yes, what will its complexity be?
Is there a way to eliminate such conflicts before computing the min-cost paths in the first place?

Answer 1

It's dynamic programming.
First reverse all the edges to make a new graph, we call it newG.
In newG, the node without parent has the value 0.
for every node which it have parents in newG, calculate it's
parent's value, then choose the minimal value parent, it must be the
part of the result.
when ask the path from the origin gtaph, the answer is the same
in the newG.(may be the edges in the answer is in reverse order).
Time O(n)