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.updatedSorry, my problem specification was ambiguous. I updated the sample problem explanation.update 2To clarify the specification, I provide mathematical formulation.Definition 1 transport mass mapTransport 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 transportationIn this problem, two graphs G1 and G2 are equivalent iff T(G1) is T(G2) the same sets.Lemma 1 equivalence relationIf G1 ~ G2 means G1 is equivalent with G2 in this sense, ~ is equivalence relation in a directed weighted graph set.Definition 3 equivalence setFor graph G, S*(G) is a set of graph such thatif a graph g in S*(G), g~G, and vice versa.Definition 3 minimal graph in terms of transportationGraph G- is called minimal for G iff the sum of G-'s node weights is minimum of all graph in S*(G)Problem statement 1What is (the most) efficient method to find G- when G is given.Problem statement 2 (added)What is the method above called generally?
Find equivalent weighted DAG sum of whose weight is minimum
algorithms;graphs