# Find equivalent weighted DAG sum of whose weight is minimum [closed]

When a given weighted directed acyclic graph represents some object flow, what is the most efficient algorithm to find the equivalent graph sum of which edges is minimum? Here, 'equivalent' means the end state of the objects is the same.

The picture below is the trivial example of this problem.

Above, 'simplest' means least weight sum.

## updated

Sorry, my problem specification was ambiguous. I updated the sample problem explanation.

## update 2

To clarify the specification, I provide mathematical formulation.

### Definition 1 transport mass map

Transport mass map T is a map between directed weighted graph to a set of pairs of node and transported volume.

for example,

### Definition 2 equivalence in terms of transportation

In this problem, two graphs G1 and G2 are equivalent iff T(G1) is T(G2) the same sets.

### Lemma 1 equivalence relation

If G1 ~ G2 means G1 is equivalent with G2 in this sense, ~ is equivalence relation in a directed weighted graph set.

### Definition 3 equivalence set

For graph G, S*(G) is a set of graph such that if a graph g in S*(G), g~G, and vice versa.

### Definition 3 minimal graph in terms of transportation

Graph G- is called minimal for G iff the sum of G-'s node weights is minimum of all graph in S*(G)

### Problem statement 1

What is (the most) efficient method to find G- when G is given.