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.
Sorry, my problem specification was ambiguous. I updated the sample problem explanation.
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.
Definition 2 equivalence in terms of transportation
In this problem, two graphs
G2 are equivalent iff
T(G2) the same sets.
Lemma 1 equivalence relation
G1 ~ G2 means
G1 is equivalent with
G2 in this sense,
~ is equivalence relation in a directed weighted graph set.
Definition 3 equivalence set
S*(G) is a set of graph such that
if a graph
g~G, and vice versa.
Definition 3 minimal graph in terms of transportation
G- is called minimal for
G iff the sum of
G-'s node weights is minimum of all graph in
Problem statement 1
What is (the most) efficient method to find
G is given.
Problem statement 2 (added)
What is the method above called generally?