What is the name of this network flow problem?

Question

Suppose you have the following network:

$s$ and $t$ are source and sink nodes. $s$ needs to send $3$ units of flow to $A$ and $2$ to $D$; $B$ needs to send $3$ units of flow to the sink, and $C$ $2$ units of flow. The edges between the blue nodes have infinite capacities and a fixed cost.

I want to find a feasible flow with minimum cost. The solution here is trivial: send $3$ units of flow from $A$ to $B$ and $2$ from $D$ to $C$. A sub optimal solution would be to send send $2$ units of flow from $A$ to $B$, $1$ from $A$ to $C$, $1$ from $D$ to $B$ and $1$ from $D$ to $C$.

My question: Is there a name for this problem? (it is not a typical minimum cost flow problem, because the costs are not linear). Is this problem related to the Steiner Tree problem (as we want to select a subset of edges among the graph)? I am pretty sure it is, see for example this article.

Answer 1

Your problem is related to the minimum edge-cost flow problem (MECF), which is a decision problem. Formally, it is defined as follows:

Def. (MECF) Given a budget $P$ and a directed graph $G = (V, E, c, p)$, where $V$ (resp. $E$) is the set of vertices (resp. edges), $c(e)$ is the capacity of an edge $e \in E$, and $p(e)$ is the cost by an edge $e$, test whether there exists a maximum $s,t$-flow with total price no greater than $P$ for a source $s$ and a sink $t$. Here, the price of a flow is the sum of the prices of the edges in the flow.